Free Math Worksheets

- State what the exponent \(2\) means. - Check your result by multiplying the two equal factors.

1. The exponent \(2\) means that \(14\) is used as a factor twice. 2. \(14^2 = 14 \times 14 = 196\).

\(14^2 = 14 \times 14 = 196\).

- Look for the pattern formed by consecutive perfect squares. - Match each position with a whole number from \(1\) through \(10\).

1. The sequence lists the perfect squares from \(1^2\) through \(10^2\). 2. The missing terms are \(4^2 = 16\), \(7^2 = 49\), and \(10^2 = 100\).

\(16\), \(49\), and \(100\).

- First compare the positive bases. - Calculate only as much as you need to confirm the order.

1. Because all bases are positive and all exponents are \(2\), the order of the squares follows the order of the bases: \(7 < 12 < 18 < 21\). 2. The values are \(7^2 = 49\), \(12^2 = 144\), \(18^2 = 324\), and \(21^2 = 441\).

\(7^2 < 12^2 < 18^2 < 21^2\), or \(49 < 144 < 324 < 441\).

- Look through familiar perfect squares. - Check your number by multiplying it by itself.

1. Find a whole number \(n\) such that \(n \times n = 169\). 2. Since \(13 \times 13 = 169\), the number is \(13\).

\(13\)

- Work on one part at a time. - Multiply each number by itself.

1. \(8^2 = 64\). 2. \(12^2 = 144\). 3. \(17^2 = 289\).

a) \(64\) b) \(144\) c) \(289\)

- First identify the base of the given perfect square. - Increase the base by \(1\), then square it.

1. \(225 = 15^2\). 2. The next whole-number base is \(16\). 3. \(16^2 = 256\).

\(256 = 16^2\); the base is \(16\).

- State what an exponent of \(2\) means. - Rewrite the power as a product of two equal factors.

1. The exponent \(2\) does not mean multiply the base by \(2\). It means use the base as a factor twice. 2. Therefore, \(18^2 = 18 \times 18 = 324\).

\(18^2 = 18 \times 18 = 324\).

- Which statement fully matches the definition of a perfect square? - A digit pattern or parity alone is not a proof.

1. A perfect square must be expressible as the product of two equal whole-number factors. 2. Statement B gives \(28 \times 28 = 784\), so it proves that \(784\) is a perfect square. Statements A and C do not prove the claim.

B) \(28 \times 28 = 784\).

- Represent the square arrangement with multiplication. - The number of rows and the number in each row are equal.

1. The square arrangement has \(13\) rows of \(13\) pavers. 2. \(13^2 = 13 \times 13 = 169\).

\(169\) pavers.

- Work with one base at a time. - Check that every number is used exactly once.

1. Square each base by multiplying it by itself. 2. \(6^2 = 36\), \(11^2 = 121\), \(13^2 = 169\), and \(15^2 = 225\).

\(6 \leftrightarrow 36\), \(11 \leftrightarrow 121\), \(13 \leftrightarrow 169\), \(15 \leftrightarrow 225\).

- Evaluate exponents before multiplication. - Perform multiplication before addition and subtraction. - When only addition and subtraction remain, work from left to right.

1. Evaluate the powers: \(10^2 = 100\) and \(10^1 = 10\). 2. Multiply: \(3 \times 100 = 300\) and \(2 \times 10 = 20\). 3. Evaluate from left to right: \(300 - 20 + 3 = 283\).

\(283\)

- Substitute the given number for the variable. - Evaluate parentheses and exponents before multiplication and subtraction. - Square the entire value inside the parentheses.

1. Substitute \(5\) for \(x\): \(2\times5^2-(5-2)^2\). 2. Evaluate inside the parentheses: \((5-2)^2=3^2=9\). 3. Evaluate the first term: \(2\times 5^2=2\times 25=50\). 4. Subtract: \(50-9=41\).

c) \(41\)

- Evaluate each power first. - Recall the roles of base and exponent. - Compare the final values after evaluating both expressions.

1. For \(A\), \(2^6=64\) and \(4^3=64\), so \(A=64-64=0\). 2. For \(B\), \(5^3=125\) and \(11^2=121\), so \(B=125-121=4\). 3. Since \(4>0\), \(B\) is greater.

\(B\) is greater. \(A=0\) and \(B=4\).

- Evaluate the powers before solving the equation. - Simplify both sides. - Use an inverse operation to isolate \(n\).

1. Evaluate the powers: \(2^5=32\) and \(5^2=25\). 2. Simplify the equation: \(n+32=137\). 3. Subtract \(32\) from both sides: \(n=105\).

\(n=105\)

- What does the exponent tell you about repeated multiplication? - How do you raise a fraction to a power? - Track decimal place value when multiplying decimals. - You can expand a power as repeated multiplication.

1. For a), \(0.4^2=0.4\times0.4=0.16\). 2. For b), \(\left(\frac{2}{3}\right)^3=\frac{2^3}{3^3}=\frac{8}{27}\). 3. For c), \(0.1^4=0.1\times0.1\times0.1\times0.1=0.0001\). 4. For d), \(\left(\frac{5}{2}\right)^2=\frac{5^2}{2^2}=\frac{25}{4}=6.25\).

a) \(0.16\) b) \(\frac{8}{27}\) c) \(0.0001\) d) \(\frac{25}{4}\), or \(6.25\)

- Evaluate a square by multiplying the base by itself. - Relate \(81\) to a power of \(3\). - Track decimal places when multiplying decimals.

1. For a), \(0.7^2=0.49\). 2. For b), \(3^4=81\), so \(\left(\frac{1}{3}\right)^4=\frac{1}{81}\). 3. For c), \(0.2^3=0.008\). 4. For d), \(1.1\times1.1=1.21\).

a) \(0.49\) b) \(4\) c) \(0.008\) d) \(1.21\)

- In Expression A, multiply before adding or subtracting. - In Expressions B and C, evaluate the parentheses first. - Pay attention to signs when comparing the results.

1. Expression A: \(\frac{1}{4}\times20-4+8=5-4+8=9\). 2. Expression B: \(\frac{1}{4}\times(20-4)+8=\frac{1}{4}\times16+8=12\). 3. Expression C: \(\frac{1}{4}\times20-(4+8)=5-12=-7\). 4. Since \(-7<9<12\), Expression C has the least value.

Expression A: \(9\) Expression B: \(12\) Expression C: \(-7\) Expression C has the least value.

- Convert the mixed number to a decimal. - Evaluate parentheses first. - Perform addition and subtraction from left to right.

1. For A, \(8\frac{1}{4}=8.25\), so \(A=25.5-(8.25+3.5)+2=25.5-11.75+2=15.75\). 2. For B, \(B=25.5-8.25+(3.5+2)=25.5-8.25+5.5=22.75\). 3. Since \(22.75>15.75\), Expression B has the greater value.

\(A=15.75\) \(B=22.75\) Expression B has the greater value.

- Evaluate the expression inside the parentheses first. - When subtraction applies to a grouped expression, each term inside changes sign if the grouping symbols are removed. - Compare each student’s steps with the original expression.

1. Evaluate the parentheses: \(7-12=-5\). 2. Substitute: \(-15-(-5)=-15+5=-10\). 3. Maya is correct. Lucas removed the parentheses without changing the signs of the terms inside. An equivalent ungrouped expression is \(-15-7+12\).

Maya is correct. Lucas should have changed the signs when removing parentheses preceded by subtraction. The correct value is \(-10\).

- Evaluate the parentheses first. - Think about how a subtraction sign affects every term in a grouped expression. - Compare the operations performed in the two expressions.

1. Expression A: \(-12-(5-13)=-12-(-8)=-4\). 2. Expression B: \((-12-5)-13=-17-13=-30\). 3. In Expression A, the subtraction sign applies to the entire difference \(5-13\). Removing the parentheses gives the equivalent expression \(-12-5+13\). In Expression B, the subtractions are performed from left to right.

a) Expression A is \(-4\), and Expression B is \(-30\). b) The parentheses in Expression A make the subtraction apply to the entire difference, changing the effect of the final term.

- Use the order of operations. - Evaluate grouping symbols first. - Write intermediate results when an expression has several steps.

1. For a), \(85-12\times5=85-60=25\). 2. For b), \(144\div12+8\times4=12+32=44\). 3. For c), \((35+45)\div(25-5)=80\div20=4\). 4. For d), evaluate inside the parentheses: \(4\times6=24\), so \(7\times(32-24)=7\times8=56\).

a) \(25\) b) \(44\) c) \(4\) d) \(56\)

- Evaluate each expression separately. - Identify which operation is performed first in each expression. - Describe how the parentheses change the dividend and divisor.

1. For A, divide before adding or subtracting: \(40-10\div2+3=40-5+3=38\). 2. For B, evaluate the parentheses first: \((40-10)\div(2+3)=30\div5=6\). 3. In A, only \(10\) is divided by \(2\). In B, the parentheses make the entire difference \(40-10\) the dividend and the entire sum \(2+3\) the divisor.

\(A=38\) \(B=6\) The parentheses change which quantities are divided.

- Evaluate the two sides separately. - Notice how the parentheses change which operation occurs first. - Compare the final values.

1. a) The left side is \(22\), and the right side is \(28\), so \(22 < 28\). 2. b) The left side is \(40 \div 8 = 5\), and the right side is \(3 + 2 = 5\), so the sides are equal. 3. c) The left side is \(42 - 5 = 37\), and the right side is \(7 \times 1 = 7\), so \(37 > 7\).

a) \(<\) b) \(=\) c) \(>\)

- Which operation comes first when an expression contains multiplication and addition or subtraction? - Evaluate the multiplication before completing the remaining operation. - Check each final value using the inverse operation.

1. Multiply first: \(7 \times 3 = 21\). Then \(21 + 29 = 50\). 2. Multiply first: \(7 \times 7 = 49\). Then \(49 - 14 = 35\). 3. Multiply first: \(7 \times 5 = 35\). Then \(35 + 45 = 80\). 4. Multiply first: \(7 \times 8 = 56\). Then \(56 - 26 = 30\).

1) \(50\) 2) \(35\) 3) \(80\) 4) \(30\)

- Evaluate the expression on each side of the circle separately. - Complete operations inside parentheses first. - When multiplication and division appear without parentheses, work from left to right.

1. a) The left side is \((24 \div 6) \times 5 = 4 \times 5 = 20\). The right side is \(2 \times 10 = 20\), so the correct symbol is \(=\). 2. b) Evaluate multiplication and division from left to right: \(8 \times 4 \div 2 = 32 \div 2 = 16\). Since \(16 < 20\), the correct symbol is \(<\). 3. c) The left side is \((4 \times 3) + (6 \times 3) = 12 + 18 = 30\). The right side is \(9 \times 3 = 27\). Since \(30 > 27\), the correct symbol is \(>\).

a) \(=\) b) \(<\) c) \(>\)

- Identify which operation must be completed first in each expression. - Recall the usual order for multiplication, division, addition, and subtraction. - Compare parts a and b, and then compare parts c and d.

1. a) Divide before adding: \(48 \div 6 + 2 = 8 + 2 = 10\). 2. b) Evaluate the parentheses first: \(48 \div (6 + 2) = 48 \div 8 = 6\). 3. c) Multiply before subtracting: \(5 \times 9 - 4 = 45 - 4 = 41\). 4. d) Evaluate the parentheses first: \(5 \times (9 - 4) = 5 \times 5 = 25\).

a) \(10\) b) \(6\) c) \(41\) d) \(25\)

- How are the side lengths of a square related? - Find a whole number that gives \(81\) when multiplied by itself. - Use known multiplication facts or perfect squares.

1. A square has the same number of tiles in each row and each column. 2. If \(\square\) represents the number of tiles on each side, the equation is \(\square \times \square = 81\). 3. Since \(9 \times 9 = 81\), there are \(9\) tiles in each row. As a square power, \(9^2 = 81\).

a) \(\square \times \square = 81\) b) \(9\) tiles; \(9^2 = 81\).

- Multiplication and division have the same priority. - What rule applies when operations have equal priority? - Compare the original expression with \(80 \div (4 \times 2)\).

1. Multiplication and division have equal priority. 2. When operations have equal priority and there are no parentheses, evaluate them from left to right. 3. \(80 \div 4 = 20\), and \(20 \times 2 = 40\). 4. Lena is correct. Jan evaluated the different expression \(80 \div (4 \times 2)\).

Lena is correct. \(80 \div 4 \times 2 = 40\).

- Identify which operations would be completed first even without parentheses. - Think about whether parentheses are needed in a chain containing only addition. - After removing a pair, check that the value remains unchanged.

1. a) Multiplication is completed before addition, so the parentheses around the products are unnecessary. The expression becomes \(12 \times 4 + 7 \times 8 = 48 + 56 = 104\). 2. b) All parentheses can be removed because the expression contains only addition. The expression becomes \(35 + 15 + 20 = 50 + 20 = 70\). 3. c) Division is completed before subtraction, so the parentheses are unnecessary. The expression becomes \(90 \div 9 - 4 = 10 - 4 = 6\).

a) \(12 \times 4 + 7 \times 8 = 104\) b) \(35 + 15 + 20 = 70\) c) \(90 \div 9 - 4 = 6\)

- Check whether the expression contains grouping symbols. - When multiplication and division occur in sequence, they are evaluated in a fixed direction. - Consider whether starting with the division on the right would preserve the expression's value.

1. Division operations have equal priority, so evaluate them from left to right. 2. First, \(80 \div 8 = 10\). 3. Then divide the intermediate result by \(2\): \(10 \div 2 = 5\).

The value is \(5\). Division is performed from left to right: \(80 \div 8 = 10\), and then \(10 \div 2 = 5\).

- The exponent tells how many equal factors are multiplied. - A product made entirely of factors equal to \(1\) has value \(1\). - For a power of \(10\), relate the exponent to the number of zeros.

1. a) \(10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100{,}000\). 2. b) \(1^{100}\) is a product of one hundred factors equal to \(1\), so its value is \(1\). 3. c) \(13^2 = 13 \times 13 = 169\). 4. d) \(2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).

a) \(10 \times 10 \times 10 \times 10 \times 10 = 100{,}000\) b) \(1^{100} = 1\), because it is a product of one hundred factors equal to \(1\). c) \(13 \times 13 = 169\) d) \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\)

- Track what happens to the number of players after each round. - Write a list of powers of \(2\). - How many factors of \(2\) make \(128\)? - In the final round, only two players remain before one champion is determined.

1. In a single-elimination tournament with a power-of-two number of players, \(2^n\) players require \(n\) rounds because the field is halved in each round. 2. a) Since \(2^7 = 128\), the tournament requires \(7\) rounds. 3. b) A tournament lasting \(9\) rounds can begin with \(2^9\) players. Since \(2^9 = 512\), the greatest possible number is \(512\).

a) \(7\) rounds, because \(2^7 = 128\). b) \(512\) players, because \(2^9 = 512\).

- A square means multiplying a number by itself. - Use place value to determine the decimal point in each product. - Compare the number of decimal places in the base with the product.

1. \(0.7^2=0.7\times0.7=0.49\). 2. \(0.12^2=0.12\times0.12=0.0144\). 3. \(0.09^2=0.09\times0.09=0.0081\). 4. \(1.3^2=1.3\times1.3=1.69\). 5. \(0.015^2=0.015\times0.015=0.000225\).

a) \(0.49\) b) \(0.0144\) c) \(0.0081\) d) \(1.69\) e) \(0.000225\)

- Use parentheses around the negative input. - Evaluate exponents before multiplying. - Remember that an even power of a negative number is positive, while an odd power is negative.

1. \(3\times(-3)+10=1\). 2. \((-3)^2-5=9-5=4\). 3. \(2\times(-3)^2+(-3)=18-3=15\). 4. \(\frac{1}{3}\times(-3)^3=\frac{1}{3}\times(-27)=-9\).

a) \(1\) b) \(4\) c) \(15\) d) \(-9\)

- Compare each number with familiar perfect squares. - When needed, check the perfect squares immediately below and above the number.

1. A perfect square can be written as the product of two equal whole-number factors. 2. \(64 = 8^2\), \(81 = 9^2\), \(121 = 11^2\), and \(196 = 14^2\). The other numbers are not perfect squares.

\(64\), \(81\), \(121\), and \(196\).

- Look for two equal factors. - Support your conclusion with an equation.

1. A perfect square can be written as the product of two equal whole-number factors. 2. Since \(50 \times 50 = 2500\), the number \(2500\) is a perfect square.

Yes. \(2500 = 50^2 = 50 \times 50\).

- Evaluate both perfect squares or use their positions in the sequence of perfect squares. - Compare the difference with the stated value.

1. \(20^2 = 400\) and \(21^2 = 441\). 2. The difference is \(441 - 400 = 41\), so Mara’s claim is correct.

The claim is correct because \(21^2 - 20^2 = 41\).

- Evaluate the two perfect squares separately. - Then determine whether their difference is even or odd.

1. \(12^2 = 144\) and \(11^2 = 121\). 2. \(144 - 121 = 23\). 3. Since \(23\) is not divisible by \(2\), it is odd.

The difference is \(23\), and it is odd.

- Find the distance from \(300\) to each perfect square. - The smaller distance identifies the closer number.

1. The distance from \(300\) to \(289\) is \(300 - 289 = 11\). 2. The distance from \(300\) to \(324\) is \(324 - 300 = 24\). 3. Since \(11 < 24\), \(289\) is closer to \(300\).

\(289\) is closer; the distances are \(11\) and \(24\).

- Find the greatest perfect square less than \(501\). - Then check the next perfect square.

1. \(22^2 = 484\). 2. The next perfect square is \(23^2 = 529\). 3. Therefore, \(484 < 501 < 529\).

\(501\) lies between \(484 = 22^2\) and \(529 = 23^2\).

- Check each statement independently. - Identify the incorrect result before changing it.

1. The first three statements are correct: \(9^2 = 81\), \(14^2 = 196\), and \(19^2 = 361\). 2. The statement \(23^2=549\) is incorrect because \(23^2=529\).

The incorrect statement is \(23^2=549\). It should be \(23^2=529\).

- Rewrite the power as multiplication. - Use that multiplication equation to write a related division equation.

1. \(12^2 = 144\) means \(12 \times 12 = 144\). 2. A related division equation is \(144 \div 12 = 12\). 3. Ben used the exponent \(2\) as the divisor, but \(144 \div 2 = 72\).

The related division equation is \(144 \div 12 = 12\), not \(144 \div 2 = 12\).

- Describe the rows and columns in the array. - Check whether the numbers of rows and columns are equal.

1. The array has \(8\) rows and \(8\) columns. 2. \(8 \times 8 = 64\). 3. Therefore, the total is \(64 = 8^2\) dots, and each side is \(8\) dots long.

The array has \(64\) dots, and each side is \(8\) dots long.

- Check each dot array separately. - For each total, look for two equal whole-number factors.

1. Array a) has \(36 = 6^2\) dots, so it represents a perfect square. 2. Array b) has \(42\) dots. Since \(6^2 = 36 < 42 < 49 = 7^2\), it does not represent a perfect square. 3. Array c) has \(49 = 7^2\) dots, so it represents a perfect square.

a) \(36 = 6^2\) c) \(49 = 7^2\)

- Evaluate each power before adding or subtracting. - In part b), focus only on the term that changes. - Think about what happens when the amount being subtracted becomes smaller.

1. Evaluate the powers: \(3^4=81\), \(2^5=32\), and \(7^2=49\). 2. For a), \(81+32-49=64\). 3. Replacing \(7^2\) with \(6^2\) changes the subtracted value from \(49\) to \(36\). 4. A number \(13\) smaller is being subtracted, so the entire expression increases by \(49-36=13\).

a) \(64\) b) The value increases by \(13\).

- Follow the order of operations: parentheses, exponents, multiplication and division, then addition and subtraction. - Convert the fraction to a decimal before adding inside the parentheses. - Squaring a number means multiplying it by itself.

1. Evaluate inside the parentheses: \(\frac{4}{5}=0.8\), so \(0.4+0.8=1.2\). 2. Evaluate the division: \(1.2\div0.5=2.4\). 3. Evaluate the exponent: \((1.2)^2=1.44\). 4. Subtract: \(2.4-1.44=0.96\).

\(0.96\)

- What does the exponent \(2\) tell you to do with the fraction? - Evaluate the power before the multiplication. - Rewrite the mixed number as an improper fraction. - Remember to write the final result as a mixed number.

1. Evaluate the exponent first: \(\left(\frac{4}{5}\right)^2=\frac{16}{25}\). 2. Rewrite \(1 \frac{7}{8}\) as \(\frac{15}{8}\). 3. Multiply and simplify: \(\frac{16}{25}\times\frac{15}{8}=\frac{6}{5}=1 \frac{1}{5}\).

\(1 \frac{1}{5}\)

- What happens to the denominator each time you multiply by \(\frac{1}{2}\)? - Write the results after one, two, and three halvings and look for a pattern. - What power of \(2\) equals \(64\)?

1. Each time you take one half, you multiply by \(\frac{1}{2}\). 2. After \(n\) halvings, the value is \(\left(\frac{1}{2}\right)^n=\frac{1}{2^n}\). 3. To get \(\frac{1}{64}\), solve \(2^n=64\). 4. Since \(2^6=64\), the value must be halved \(6\) times.

6 times, because \(\left(\frac{1}{2}\right)^6=\frac{1}{64}\).

- Mark each multiplication and division before doing any addition or subtraction. - Work from the innermost grouping symbols outward. - Pay close attention to signs when multiplying, dividing, or subtracting negative numbers.

1. For a), evaluate multiplication and division first: \((-32)\div4=-8\) and \((-12)\times2=-24\). Then \(-8-(-24)=16\). 2. For b), evaluate the brackets first: \(40-(-10)=50\). Then \(150-50\times3=150-150=0\). 3. For c), evaluate multiplication and division first: \((-0.2)\times50=-10\) and \((-10)\div(-2)=5\). Then \(-10+5=-5\).

a) \(16\) b) \(0\) c) \(-5\)

- Consider which adjacent part should be grouped to change the order of operations. - After evaluating the parentheses, multiply and divide from left to right. - Check the sign of the final product.

1. Put the parentheses around \(2+3\): \(-20\div(2+3)\times(-2)\). 2. Evaluate the parentheses: \(2+3=5\). 3. Multiply and divide from left to right: \(-20\div5\times(-2)=-4\times(-2)=8\).

\(-20\div(2+3)\times(-2)=8\)

- Exponents are evaluated before subtraction. - Test whether \((a-b)^2=a^2-b^2\) using simple numbers. - Evaluate each square separately before subtracting.

1. The error is combining the bases before evaluating the exponents. In general, \(a^2-b^2\ne(a-b)^2\). 2. Evaluate each power separately: \((-0.3)^2=0.09\) and \(0.1^2=0.01\). 3. Subtract: \(0.09-0.01=0.08\).

The step \((-0.3)^2-0.1^2=(-0.3-0.1)^2\) is incorrect. The correct value is \(0.08\).

- Evaluate the power on the left side first. - What happens when a positive number less than \(1\) is multiplied by itself? - Converting between fractions and decimals can make comparisons easier.

1. For a), \(0.2^2=0.04\), and \(0.04<0.2\). 2. For b), \(\left(\frac{1}{2}\right)^3=\frac{1}{8}\) and \(\left(\frac{1}{2}\right)^2=\frac{1}{4}\), so \(\frac{1}{8}<\frac{1}{4}\). 3. For c), \(0.1^2=0.01\), so the two values are equal. 4. For d), \(\left(\frac{3}{2}\right)^2=\frac{9}{4}=2.25\), and \(2.25>1.5\).

a) \(<\) b) \(<\) c) \(=\) d) \(>\)

- Consider what the order of operations would require without parentheses. - A subtraction sign before parentheses affects every term inside. - Compare what happens when parentheses follow addition versus subtraction.

1. For a), the parentheses can be removed because the grouped difference is being added: \(24.5+12.8-5.2\) has the same value. 2. For b), the parentheses cannot be removed. Without them, only \(6.4\times3\) would be evaluated before the subtraction. 3. For c), the parentheses cannot simply be removed. The subtraction applies to the entire sum, so removing the parentheses would require changing both signs: \(40-10.5-4.5\).

a) Yes b) No c) No

- Evaluate the grouped operation first. - When multiplication and division appear without parentheses, work from left to right. - Compare the two results for each part.

1. For a), with parentheses: \((3.6\times5)\div2=18\div2=9\). Without parentheses, multiply and divide from left to right: \(3.6\times5\div2=18\div2=9\). The parentheses do not change the value. 2. For b), with parentheses: \(36\div(3\times2)=36\div6=6\). Without parentheses, multiply and divide from left to right: \(36\div3\times2=12\times2=24\). The parentheses change the value.

a) The parentheses do not change the value; both expressions equal \(9\). b) The parentheses change the value; the expression equals \(6\) with them and \(24\) without them.

- Check whether removing grouping symbols changes the order of operations. - Consider whether the negative factor remains clearly written. - Pay special attention to the subtraction sign before the last parentheses.

1. The parentheses around \(-12\) can be removed: \([-12+4]\) is clear and has the same value. 2. The brackets cannot be removed. Without them, \(4\times(-3)\) would be evaluated before the addition. 3. Keep the parentheses around \(-3\). They clearly show that the negative number is a factor and avoid placing an operation sign directly next to a negative sign. 4. The parentheses around \(10-2\) cannot be removed. The subtraction outside the parentheses applies to the entire difference.

1. Yes 2. No 3. No 4. No

- Find the total elevation gain first. - Determine how many \(1000\,\text{ft}\) intervals are in that gain. - The temperature decreases as the group climbs.

1. Find the elevation gain: \(9000-2000=7000\,\text{ft}\). 2. Find the number of \(1000\,\text{ft}\) intervals: \(7000\div1000=7\). 3. Find the temperature drop: \(7\times3.5=24.5^\circ\text{F}\). 4. Subtract from the starting temperature: \(68-24.5=43.5^\circ\text{F}\). One expression is \(68-(9000-2000)\times\frac{3.5}{1000}\).

The expected temperature is \(43.5^\circ\text{F}\). One suitable expression is \(68-(9000-2000)\times\frac{3.5}{1000}\).

- Try grouping the first two numbers, the last two numbers, or a middle part of the expression. - Evaluate each attempt using the order of operations. - Decide which operation must occur first to make the result larger or smaller.

1. For a), group the first subtraction: \((15-3)\times2.5+0.5=12\times2.5+0.5=30.5\). 2. For b), group the final addition: \(15-3\times(2.5+0.5)=15-3\times3=6\).

a) \((15-3)\times2.5+0.5=30.5\) b) \(15-3\times(2.5+0.5)=6\)

- Convert the mixed numbers to decimals before testing placements. - Consider how a subtraction sign before parentheses changes the result. - Try different contiguous groups and evaluate each one.

1. Convert the mixed numbers: \(4\frac{1}{2}=4.5\) and \(2\frac{1}{5}=2.2\). 2. For a), group the final sum: \(12.8-4.5-(1.3+2.2)=8.3-3.5=4.8\). 3. For b), group all terms after the first subtraction sign: \(12.8-(4.5-1.3+2.2)=12.8-5.4=7.4\).

a) \(12.8-4\frac{1}{2}-(1.3+2\frac{1}{5})=4.8\) b) \(12.8-(4\frac{1}{2}-1.3+2\frac{1}{5})=7.4\)

- Evaluate the expression inside the parentheses before subtracting it. - Without parentheses, add and subtract from left to right. - Compare the two final values.

1. For a), evaluate inside the parentheses: \(7.5-2.4+1.1=6.2\). Then \(30-6.2=23.8\). 2. For b), work from left to right: \(30-7.5-2.4+1.1=22.5-2.4+1.1=21.2\). 3. Jordan is not correct because \(23.8\ne21.2\). The subtraction sign before the parentheses applies to the entire grouped expression.

a) \(23.8\) b) \(21.2\) c) No; the two expressions have different values.

- Without parentheses, perform multiplication and division before subtraction. - When multiplication and division occur in sequence, work from left to right. - Compare negative values carefully.

1. Expression A: evaluate division and multiplication first. \(-2.4\div0.6=-4\) and \(0.2\times5=1\), so the value is \(-4-1=-5\). 2. Expression B: evaluate the parentheses first. \(0.6-0.2=0.4\). Then multiply and divide from left to right: \(-2.4\div0.4\times5=-6\times5=-30\). 3. Since \(-30<-5\), Expression B has the lesser value.

Expression A: \(-5\) Expression B: \(-30\) Expression B has the lesser value.

- Notice how the parentheses change the order of operations. - Convert the fraction to a decimal if helpful. - In Expression B, divide before subtracting.

1. For A, evaluate the parentheses first: \(2.4-\frac{2}{5}=2.4-0.4=2\). Then \(2\div0.5=4\). 2. For B, divide before subtracting: \(\frac{2}{5}\div0.5=0.4\div0.5=0.8\). Then \(2.4-0.8=1.6\). 3. Since \(4>1.6\), Expression A has the greater value.

\(A=4\) \(B=1.6\) Expression A has the greater value.

- Evaluate parentheses and exponents before multiplication, division, addition, or subtraction. - The square of a negative number is positive. - Rewrite division by a fraction as multiplication by its reciprocal.

1. For a), evaluate the parentheses: \(3.5+(-1.2)=2.3\). Then \(2.4-2.3=0.1\). 2. For b), evaluate the exponent and multiplication: \((-0.5)^2=0.25\) and \(\frac{3}{4}\times2=1.5\). Then \(0.25-1.5=-1.25\). 3. For c), evaluate the parentheses: \(0.2-0.5=-0.3=-\frac{3}{10}\). Then \(\frac{2}{5}\div\left(-\frac{3}{10}\right)=\frac{2}{5}\times\left(-\frac{10}{3}\right)=-\frac{4}{3}\).

a) \(0.1\) b) \(-1.25\) c) \(-\frac{4}{3}\)

- Test grouping the first two numbers and then the last two numbers. - Parentheses can make addition or subtraction occur before division. - Evaluate each candidate carefully.

1. Without added parentheses, divide first: \(12+48\div6-2=12+8-2=18\). 2. For a value of \(8\), use \((12+48)\div6-2=60\div6-2=8\). 3. For a value of \(24\), use \(12+48\div(6-2)=12+48\div4=24\).

a) \(18\) b) \((12+48)\div6-2=8\) c) \(12+48\div(6-2)=24\)

- Evaluate parentheses first. - The order of operations also applies inside parentheses. - Work outward one operation at a time.

1. a) \(16 + 24 = 40\), \(40 \div 8 = 5\), and \(5 + 9 = 14\). 2. b) Inside the parentheses, multiply first: \(5 \times 6 - 12 = 30 - 12 = 18\). Then \(18 \div 3 = 6\). 3. c) \(45 + 15 = 60\), \(60 \div 10 = 6\), and \(6 + 24 = 30\).

a) \(14\) b) \(6\) c) \(30\)

- Evaluate A and B separately. - Record the value of each expression in parentheses. - Compare the two final values.

1. For A, \(7 \times 8 + 4 = 56 + 4 = 60\), so \(60 \div 6 + 30 = 10 + 30 = 40\). 2. For B, \(3 \times 9 + 3 = 27 + 3 = 30\), so \(100 - 30 \div 3 = 100 - 10 = 90\). 3. Since \(40 < 90\), \(A < B\).

\(A < B\), because \(40 < 90\).

- First find the value without parentheses. - Try grouping an addition or subtraction to change which operation occurs first. - Check each completed equation.

1. a) Group the subtraction: \(2 \times (10 - 5) = 2 \times 5 = 10\). 2. b) Group the addition: \(45 \div (5 + 4) = 45 \div 9 = 5\). 3. c) Group the subtraction: \((18 - 6) \times 2 = 12 \times 2 = 24\).

a) \(2 \times (10 - 5) = 10\) b) \(45 \div (5 + 4) = 5\) c) \((18 - 6) \times 2 = 24\)

- Evaluate parentheses first. - Complete multiplication and division before addition and subtraction. - For a non-exact division, identify both the quotient and remainder.

1. a) \(100 - 19 = 81\), and \(81 \div 9 = 9\). 2. b) \(60 - 15 = 45\), and \(45 \div 7 = 6\text{ R }3\). 3. c) \(5 \times 8 = 40\), then \(40 + 360 - 140 = 260\). 4. d) \(7 \times 6 = 42\), then \(42 + 118 - 50 = 110\).

a) \(9\) b) \(6\text{ R }3\) c) \(260\) d) \(110\)

- Evaluate \(10^4\) before completing the arithmetic. - Use grouping symbols for every stated sum or difference. - Translate the order of the subtractions carefully.

1. For a), \((245 + 355) - (100 - 40) = 600 - 60 = 540\). 2. For b), \(10^4 = 10{,}000\). The expression is \(10^4 - (1200 + 800) - (50 - 20)\), so its value is \(10{,}000 - 2000 - 30 = 7970\).

a) \((245 + 355) - (100 - 40) = 540\) b) \(10^4 - (1200 + 800) - (50 - 20) = 7970\)

- Evaluate the parentheses in A first. - Without parentheses, consecutive subtractions are evaluated from left to right. - Compare the two final values.

1. For A, evaluate the parentheses: \(600 - 200 = 400\), so \(1800 - 400 = 1400\). 2. For B, evaluate from left to right: \(1800 - 600 = 1200\), then \(1200 - 200 = 1000\). 3. Therefore, \(A = 1400\) and \(B = 1000\), so A is \(400\) greater than B.

\(A = 1400\) and \(B = 1000\); therefore, \(A > B\).

- Evaluate \(10^5\) first. - Use compatible rounded values for the estimate. - For the exact value, work from the inner grouping symbols outward.

1. Since \(10^5 = 100{,}000\), one estimate is \(100{,}000 - [(50{,}000 - 10{,}000) + (20{,}000 - 7000)] = 47{,}000\). 2. Evaluate exactly: \(52{,}340 - 12{,}140 = 40{,}200\) and \(18{,}500 - 7200 = 11{,}300\). 3. The bracketed sum is \(51{,}500\), so the exact value is \(100{,}000 - 51{,}500 = 48{,}500\).

Estimate: \(47{,}000\) Exact value: \(48{,}500\)

- Evaluate the operations inside each pair of parentheses first. - Then add or subtract the two intermediate values.

1. a) \(5 \times 8 = 40\) and \(3 \times 7 = 21\), so the value is \(40 + 21 = 61\). 2. b) \(9 \times 4 = 36\) and \(2 \times 8 = 16\), so the value is \(36 - 16 = 20\). 3. c) \(6 \times 6 = 36\) and \(15 \div 3 = 5\), so the value is \(36 + 5 = 41\). 4. d) \(7 \times 3 = 21\) and \(12 \div 4 = 3\), so the value is \(21 - 3 = 18\).

a) \(61\) b) \(20\) c) \(41\) d) \(18\)

- Evaluate the left and right sides separately. - Compare the two final values in each part.

1. a) \(6 \times 4 - 9 = 24 - 9 = 15\), so the correct symbol is \(=\). 2. b) \(48 \div 8 + 15 = 6 + 15 = 21\), while \(3 \times 6 = 18\). Therefore, the correct symbol is \(>\). 3. c) \(7 \times 7 - 4 \times 8 = 49 - 32 = 17\), while \(3 \times 9 + 2 = 27 + 2 = 29\). Therefore, the correct symbol is \(<\).

a) \(=\) b) \(>\) c) \(<\)

- A quantity in parentheses immediately after a minus sign is subtracted as a whole. - Try to make that grouped quantity as large as possible. - Compare several possible placements.

1. To make the value as small as possible, make the quantity subtracted from \(180\) as large as possible. 2. Group all three following numbers: \(180 - (50 + 40 + 20)\). 3. \(50 + 40 + 20 = 110\), so the minimum value is \(180 - 110 = 70\).

\(180 - (50 + 40 + 20) = 70\)

- The entire parenthetical expression after the first minus sign will be subtracted. - Make that parenthetical value as small as possible. - Test the possible placements systematically.

1. To maximize the expression, make the entire quantity subtracted from \(450\) as small as possible. 2. Group \(150 - 80 - 20\): \(450 - (150 - 80 - 20)\). 3. Inside the parentheses, evaluate from left to right: \(150 - 80 - 20 = 70 - 20 = 50\). 4. The maximum value is \(450 - 50 = 400\).

\(450 - (150 - 80 - 20) = 400\)

- First evaluate the expression without parentheses. - Decide whether the value must increase or decrease. - Test groupings that begin after the first minus sign.

1. Without parentheses, the value is \(40\), so the grouping must change the order. 2. Group the final three terms after \(75\): \(75 - (25 - 15 + 5)\). 3. Evaluate the parentheses from left to right: \(25 - 15 + 5 = 15\). 4. Then \(75 - 15 = 60\).

\(75 - (25 - 15 + 5) = 60\)

- Identify grouping symbols that do not affect the order. - Work from the innermost grouping symbols outward. - Keep the subtraction before the brackets intact.

1. The first parentheses do not change the order, so the expression can be written as \(2450 + 1550 - [1800 - (450 + 350)]\). 2. Evaluate the innermost parentheses: \(450 + 350 = 800\). 3. Evaluate the brackets: \(1800 - 800 = 1000\). 4. Evaluate the first sum: \(2450 + 1550 = 4000\). 5. Finally, \(4000 - 1000 = 3000\).

\(3000\)

- Parentheses after a plus sign are often removable. - Keep grouping that follows a minus sign when it changes the order. - Evaluate the remaining parentheses before working left to right.

1. The outer brackets and the parentheses after the plus sign are unnecessary, so write \(7600 - (2300 - 400) + 1200 + 800\). 2. Evaluate the remaining parentheses: \(2300 - 400 = 1900\). 3. Then \(7600 - 1900 = 5700\), \(5700 + 1200 = 6900\), and \(6900 + 800 = 7700\).

\(7700\)

- Examine which grouping symbols actually change the order. - Work from the innermost grouping symbols outward. - Keep the brackets because the whole bracketed value is subtracted.

1. The first parentheses inside the brackets are unnecessary, so write \(5432 - [875 + 125 - (450 - 150)]\). 2. Evaluate the inner parentheses: \(450 - 150 = 300\). 3. Evaluate the brackets: \(875 + 125 - 300 = 1000 - 300 = 700\). 4. Finally, \(5432 - 700 = 4732\).

\(4732\)

- Follow each instruction in its stated order. - Record the two intermediate and final results. - Subtract the smaller final result from the larger one.

1. Instruction A gives \(14 \times 2 + 10 = 28 + 10 = 38\). 2. Instruction B gives \((14 + 10) \times 2 = 24 \times 2 = 48\). 3. The results differ by \(48 - 38 = 10\).

Instruction A gives \(38\), instruction B gives \(48\), and the difference is \(10\).

- Evaluate the parentheses first. - The order of operations applies inside the parentheses. - Then complete the multiplication outside.

1. a) \(12 \div 3 = 4\), then \(4 \times 6 = 24\). 2. b) \(42 \div 6 = 7\), \(7 - 3 = 4\), then \(4 \times 9 = 36\). 3. c) \(56 - 49 = 7\), then \(7 \times 8 = 56\). 4. d) \(24 \div 4 = 6\), then \(6 \times 5 = 30\).

a) \(24\) b) \(36\) c) \(56\) d) \(30\)

- Evaluate one expression at a time. - Apply the order of operations inside each pair of parentheses. - Compare the four final values.

1. a) \(100 - 73 = 27\), then \(27 \div 3 \times 8 = 9 \times 8 = 72\). 2. b) \(4 \times 8 + 3 = 35\), then \(35 \div 7 \times 9 = 5 \times 9 = 45\). 3. c) \(18 \div 3 + 14 = 20\), then \(20 \div 5 \times 6 = 4 \times 6 = 24\). 4. d) \(63 \div 9 + 1 = 8\) and \(48 \div 6 = 8\), so the value is \(8 \times 8 = 64\). 5. Since \(72 > 64 > 45 > 24\), part a has the greatest value.

a) \(72\) b) \(45\) c) \(24\) d) \(64\) The greatest value is \(72\) in part a.

- Evaluate every pair of parentheses first. - Pay attention to the operation between the two parenthetical expressions. - Use multiplication and division facts you know to simplify each expression.

1. Evaluate the parentheses: \(72 \div 8 = 9\) and \(32 \div 4 = 8\). Then \(9 \times 8 = 72\). 2. Evaluate the parentheses: \(48 \div 6 = 8\) and \(7 \times 4 = 28\). Then \(8 + 28 = 36\). 3. Evaluate the parentheses: \(9 \times 9 = 81\) and \(8 \times 8 = 64\). Then \(81 - 64 = 17\). 4. Evaluate the parentheses: \(56 \div 7 = 8\) and \(45 \div 5 = 9\). Then \(8 \times 9 = 72\).

1) \(72\) 2) \(36\) 3) \(17\) 4) \(72\)

- Evaluate the multiplication in each pair of parentheses before subtracting. - Record the two products for each expression. - Compare the three final values.

1. a) \(6 \times 9 = 54\) and \(4 \times 8 = 32\), so \(54 - 32 = 22\). 2. b) \(7 \times 7 = 49\) and \(3 \times 9 = 27\), so \(49 - 27 = 22\). 3. c) \(8 \times 5 = 40\) and \(2 \times 9 = 18\), so \(40 - 18 = 22\). 4. All three expressions have the same value, \(22\), so no one result is greater than the others.

a) \(22\) b) \(22\) c) \(22\) All three results are equal.

- Evaluate the left side and the right side separately. - Complete each operation inside parentheses first. - Compare the two final values in each part.

1. a) The left side is \(8 \times 3 = 24\). The right side is also \(8 \times 3 = 24\), so \(24 = 24\). 2. b) The left side is \(6 \times 7 = 42\). The right side is \(8 \times 5 = 40\), so \(42 > 40\). 3. c) The left side is \(5 \times 6 = 30\). The right side is \(4 \times 7 = 28\), so \(30 > 28\).

a) \(=\) b) \(>\) c) \(>\)

- Use nearby hundreds or thousands that are easy to compute mentally for the estimate. - For the exact value, work from the innermost grouping symbols outward. - For each pair of grouping symbols, check whether removing it would change the order of operations.

1. One reasonable estimate is \(7000 - [2000 + (1000 - 400)] = 7000 - 2600 = 4400\). 2. Evaluate the parentheses: \(960 - 440 = 520\). 3. Evaluate the square brackets: \(2140 + 520 = 2660\). 4. Subtract: \(6820 - 2660 = 4160\). 5. The parentheses can be removed because \(2140 + 960 - 440\) is evaluated from left to right and still equals \(2660\). The square brackets cannot be removed because the subtraction outside them applies to the entire sum inside.

Estimate: \(\approx 4400\) Exact value: \(4160\) The parentheses can be removed, but the square brackets are necessary.

- Recall how an expression containing only addition and subtraction is evaluated. - Without grouping symbols, addition and subtraction are performed from left to right. - Mentally test the expression after removing each pair of grouping symbols.

1. One reasonable estimate is \(5400 - 1200 - 800 + 1600 = 5000\). 2. Evaluate the parentheses: \(5410 - 1190 = 4220\). 3. Continue inside the square brackets: \(4220 - 820 = 3400\). 4. Add: \(3400 + 1550 = 4950\). 5. The parentheses are unnecessary because subtraction is already performed from left to right. The square brackets are also unnecessary because the expression outside them adds \(1550\), so removing the brackets does not change the operation order.

Estimate: \(\approx 5000\) Exact value: \(4950\) Both the parentheses and the square brackets can be removed.

- Think about how parentheses can change the usual order of operations. - In part a, the quotient must be small. Consider how to make the divisor larger. - In part b, test what happens when the addition is completed before the surrounding operations.

1. a) Group the entire divisor: \(48 \div (6 + 2 \times 5) = 48 \div (6 + 10) = 48 \div 16 = 3\). 2. b) Group \(6 + 2\), then evaluate multiplication and division from left to right: \(48 \div (6 + 2) \times 5 = 48 \div 8 \times 5 = 6 \times 5 = 30\).

a) \(48 \div (6 + 2 \times 5) = 3\) b) \(48 \div (6 + 2) \times 5 = 30\)

- Test the possible locations for one pair of parentheses. - Consider whether grouping terms before multiplying by \(5\) can increase the value. - Evaluate and compare each distinct result.

1. Check the meaningful ways to change the grouping. 2. Grouping \(12 - 4 + 6\) gives \(5 \times (12 - 4 + 6) = 5 \times 14 = 70\). 3. Grouping only \(12 - 4\) gives \(5 \times (12 - 4) + 6 = 5 \times 8 + 6 = 46\). 4. Grouping \(4 + 6\) gives \(5 \times 12 - (4 + 6) = 60 - 10 = 50\). Groupings that preserve the original operation order give \(62\). 5. Since \(70\) is greater than \(62\), \(50\), and \(46\), the greatest possible value is \(70\).

The greatest possible value is \(70\), produced by \(5 \times (12 - 4 + 6)\).

- Evaluate the left and right sides separately. - Complete the operations inside parentheses first. - Compare the two final values in each part.

1. a) The left side is \((72 - 63) \times 8 = 9 \times 8 = 72\). The right side is \((100 - 28) \div 9 = 72 \div 9 = 8\). Therefore, \(72 > 8\). 2. b) The left side is \((45 - 38) \times 6 = 7 \times 6 = 42\). The right side is \((91 - 56) \div 5 = 35 \div 5 = 7\). Therefore, \(42 > 7\). 3. c) The left side is \((100 - 19) \div 9 = 81 \div 9 = 9\). The right side is \((34 - 25) \times 1 = 9 \times 1 = 9\). Therefore, \(9 = 9\).

a) \(>\) b) \(>\) c) \(=\)

- Look for factors you can combine to make the calculation easier. - Parentheses determine which product must be found first. - In part a, consider dividing \(54\) by \(9\) before multiplying.

1. a) Divide a convenient factor first: \(42 \times (54 \div 9) = 42 \times 6 = 252\). 2. b) Regroup convenient factors: \(15 \times 12 \times 5 = 15 \times (12 \times 5) = 15 \times 60 = 900\). 3. c) Evaluate the parentheses first: \(5 \times 20 = 100\), so \(800 \div 100 = 8\). 4. d) Regroup convenient factors: \(12 \times 11 \times 5 = (12 \times 5) \times 11 = 60 \times 11 = 660\).

a) \(252\) b) \(900\) c) \(8\) d) \(660\)

- Multiplication is completed before addition and subtraction. - After multiplication, evaluate addition and subtraction from left to right. - Insert parentheses to show how Lukas and Paul obtained their results.

1. Lukas calculated from left to right without doing multiplication first: \((100 - 10) \times 5 + 5 = 455\). 2. Mia followed the order of operations: \(100 - (10 \times 5) + 5 = 100 - 50 + 5 = 55\). 3. Paul effectively added the final two \(5\)s first: \(100 - 10 \times (5 + 5) = 0\), which changes the original expression. 4. Mia is correct.

Lukas worked left to right without multiplying first. Mia multiplied first and obtained the correct value, \(55\). Paul grouped the last two \(5\)s, which changed the expression.

- Test what happens when the first addition is grouped. - Test what happens when the subtraction at the end is grouped. - For part c, apply the order of operations without adding grouping symbols.

1. a) Group the first sum: \((12 + 8) \times 5 - 3 = 20 \times 5 - 3 = 100 - 3 = 97\). 2. b) Group the final difference: \(12 + 8 \times (5 - 3) = 12 + 8 \times 2 = 12 + 16 = 28\). 3. c) Without added parentheses, multiply first: \(12 + 8 \times 5 - 3 = 12 + 40 - 3 = 49\).

a) \((12 + 8) \times 5 - 3 = 97\) b) \(12 + 8 \times (5 - 3) = 28\) c) \(49\)

- Try grouping different parts of the expression. - Decide which operation must be completed first to reach each target value. - Remember that parentheses can change the usual order of operations.

1. a) Group the first difference: \((18 - 6) \div 3 + 3 = 12 \div 3 + 3 = 4 + 3 = 7\). 2. b) Group the final sum: \(18 - 6 \div (3 + 3) = 18 - 6 \div 6 = 18 - 1 = 17\). 3. c) Group both the first difference and the final sum: \((18 - 6) \div (3 + 3) = 12 \div 6 = 2\).

a) \((18 - 6) \div 3 + 3 = 7\) b) \(18 - 6 \div (3 + 3) = 17\) c) \((18 - 6) \div (3 + 3) = 2\)

- Use the order of operations to decide which grouping symbols affect the value. - Operations of equal priority are evaluated from left to right. - Square brackets and parentheses both indicate grouping.

1. a) The square brackets are unnecessary, but the parentheses around \(12 + 8\) must remain. The simplified expression is \(100 - (12 + 8) \times 3 = 100 - 20 \times 3 = 40\). 2. b) The parentheses around \(15 \times 4\) and the square brackets are unnecessary. The parentheses around \(10 - 7\) must remain. The simplified expression is \(15 \times 4 \times 2 \times (10 - 7) = 60 \times 2 \times 3 = 360\). 3. c) The parentheses around \(48 \div 6\) are unnecessary, but the grouping around the sum must remain. The simplified expression is \((48 \div 6 + 12) \times 2 = (8 + 12) \times 2 = 40\).

a) \(100 - (12 + 8) \times 3 = 40\) b) \(15 \times 4 \times 2 \times (10 - 7) = 360\) c) \((48 \div 6 + 12) \times 2 = 40\)

- Apply the order of operations to part a. - Consider how grouping an addition or subtraction before the multiplication changes the value. - Test one possible pair of parentheses at a time.

1. a) Multiply first, then add and subtract from left to right: \(80 - 4 \times 12 + 3 = 80 - 48 + 3 = 35\). 2. b) Group the final sum: \(80 - 4 \times (12 + 3) = 80 - 4 \times 15 = 80 - 60 = 20\). 3. c) Group the first difference: \((80 - 4) \times 12 + 3 = 76 \times 12 + 3 = 912 + 3 = 915\).

a) \(35\) b) \(80 - 4 \times (12 + 3) = 20\) c) \((80 - 4) \times 12 + 3 = 915\)

- First evaluate each expression using the usual order of operations. - Work backward from the target value to identify a useful intermediate result. - Consider grouping an addition or subtraction so it is completed before multiplication.

1. a) Group the sum: \((8 + 2) \times 11 = 10 \times 11 = 110\). 2. b) Group the difference: \((25 - 5) \times 3 = 20 \times 3 = 60\). 3. c) Group the final difference: \(12 \times (10 - 5) = 12 \times 5 = 60\).

a) \((8 + 2) \times 11 = 110\) b) \((25 - 5) \times 3 = 60\) c) \(12 \times (10 - 5) = 60\)

- Use the target value to think about a helpful intermediate result. - Parentheses can make an addition or subtraction occur before multiplication. - Working backward from the final value may help identify the needed grouping.

1. a) Group \(14 - 4\): \(5 \times (14 - 4) + 12 = 5 \times 10 + 12 = 62\). 2. b) Group \(8 - 5\): \(20 + 10 \times (8 - 5) = 20 + 10 \times 3 = 50\). 3. c) Group \(6 + 4\): \(44 - 4 \times (6 + 4) = 44 - 4 \times 10 = 4\).

a) \(5 \times (14 - 4) + 12 = 62\) b) \(20 + 10 \times (8 - 5) = 50\) c) \(44 - 4 \times (6 + 4) = 4\)

- Think about how division can produce \(1\). - Remember that grouping symbols are evaluated first. - You may join digits to form a larger whole number. - Combine small intermediate results with addition or subtraction.

1. A value of \(1\) can be made by dividing equal numbers: \(3 \div 3 = 1\). 2. A value of \(2\) can be made with \((3 + 3) \div 3 = 2\). 3. A value of \(4\) can be made with \(3 + 3 \div 3 = 4\). 4. A value of \(5\) can be made with \(3 + (3 + 3) \div 3 = 5\). 5. Using exactly three digits, \(33 - 3 = 30\).

Possible answers: a) \(1 = 3 \div 3\); \(2 = (3 + 3) \div 3\); \(4 = 3 + 3 \div 3\); \(5 = 3 + (3 + 3) \div 3\) b) \(30 = 33 - 3\)

- Use all four digits in every part. - Think about how equal values can produce \(0\) or \(1\). - Apply the order of operations carefully. - Grouping symbols may help change which operation is completed first.

1. a) Subtract equal values: \(4 + 4 - 4 - 4 = 0\). 2. b) Divide equal values: \((4 + 4) \div (4 + 4) = 1\). 3. c) Add three \(4\)s and subtract one: \(4 + 4 + 4 - 4 = 8\). 4. d) Apply multiplication and division before subtraction: \(4 \times 4 - 4 \div 4 = 16 - 1 = 15\).

Possible answers: a) \(4 + 4 - 4 - 4 = 0\) b) \((4 + 4) \div (4 + 4) = 1\) c) \(4 + 4 + 4 - 4 = 8\) d) \(4 \times 4 - 4 \div 4 = 15\)

- Think of factor pairs whose product is \(100\). - Consider whether division can produce \(11\). - Joining two digits to form \(55\) may be useful.

1. One expression for \(100\) is \((5 + 5 + 5 + 5) \times 5 = 20 \times 5 = 100\). 2. A different expression is \(5 \times (5 \times 5 - 5) = 5 \times 20 = 100\). 3. Using exactly three digits, \(55 \div 5 = 11\).

Possible answers: a) \((5 + 5 + 5 + 5) \times 5 = 100\) and \(5 \times (5 \times 5 - 5) = 100\) b) \(55 \div 5 = 11\)

- Identify which operations have priority. - When only addition and subtraction remain, evaluate from left to right. - Reconstruct the incorrect order Max used.

1. a) Max added before multiplying. Multiplication must be completed first: \(12 + 3 \times 5 = 12 + 15 = 27\). 2. b) Max grouped \(50 + 10\) and subtracted the sum. Addition and subtraction have equal priority, so evaluate from left to right: \(100 - 50 + 10 = 50 + 10 = 60\).

a) Max ignored multiplication before addition. The correct result is \(27\). b) Max did not evaluate from left to right. The correct result is \(60\).

- Evaluate the original parentheses first. - After that, which operation has priority? - Check whether Mia introduced parentheses that were not in the original expression.

1. Mia correctly evaluates the parentheses: \(15 + 5 = 20\). 2. She then incorrectly subtracts \(2\) from \(4\) before multiplying. The original expression does not contain parentheses around \(4 - 2\). 3. Follow the order of operations: \(20 \times 4 - 2 = 80 - 2 = 78\).

Mia incorrectly evaluated \(4 - 2\) before multiplication. The correct value is \(78\).

- Think about how grouping can enlarge the factors in a product. - To make the result small, try subtracting the largest possible product. - Check every allowed grouping against the order of operations.

1. To obtain the greatest value, make both factors as large as possible: \((15-3)\times(4+6)=12\times10=120\). 2. To obtain the least value, make the quantity subtracted from \(15\) as large as possible: \(15-3\times(4+6)=15-30=-15\).

Greatest value: \((15-3)\times(4+6)=120\) Least value: \(15-3\times(4+6)=-15\)

- Test what happens when the addition is completed before multiplication. - Test what happens when the final subtraction is completed before multiplication. - Compare all distinct results.

1. To maximize the value, make the first sum a factor: \((8 + 2) \times 10 - 5 = 10 \times 10 - 5 = 95\). 2. To minimize the value, group the final difference: \(8 + 2 \times (10 - 5) = 8 + 2 \times 5 = 18\). 3. Checking all distinct full groupings gives possible values of \(18\), \(23\), \(50\), and \(95\). Therefore, \(95\) is greatest and \(18\) is least.

Greatest value: \((8 + 2) \times 10 - 5 = 95\) Least value: \(8 + 2 \times (10 - 5) = 18\)

- Evaluate exponents before the other operations. - Start with the innermost grouping symbols. - Apply multiplication before addition or subtraction within the same grouping level. - A difference can be negative when the number being subtracted is larger.

1. Evaluate the exponent: \(3^2=9\). 2. Evaluate the parentheses: \(9+11=20\). 3. Continue inside the brackets: \(20\times2=40\). 4. Finish the brackets: \(40-10=30\). 5. Multiply before subtracting in the main expression: \(3\times30=90\). 6. Finally, \(60-90=-30\).

\(-30\)

- Decide whether multiplication is needed to reach each target value. - Apply multiplication before addition or subtraction unless parentheses change the order. - If the usual order does not work, try grouping an addition or subtraction. - Test the possible operation combinations systematically.

1. a) Multiply, then add: \(7 \times 2 + 5 = 14 + 5 = 19\). 2. b) Group the sum before multiplying: \((9 + 4) \times 2 = 13 \times 2 = 26\). 3. c) Multiply, then add: \(20 + 5 \times 3 = 20 + 15 = 35\).

a) \(7 \times 2 + 5 = 19\) b) \((9 + 4) \times 2 = 26\) c) \(20 + 5 \times 3 = 35\)

- Addition and subtraction have equal priority. - Evaluate the original expression from left to right. - Compare it with \(100 - (40 - 10)\).

1. Consecutive addition and subtraction operations have equal priority and are evaluated from left to right. 2. \(100 - 40 = 60\), then \(60 - 10 = 50\). 3. The student effectively inserted parentheses to create \(100 - (40 - 10)\), which is not the original expression.

The claim is incorrect. \(100 - 40 - 10 = 50\).

- How do parentheses affect the order of operations? - Evaluate the quantity inside the parentheses first in Expression B. - In which direction is Expression A evaluated when there are no grouping symbols?

1. For Expression A, evaluate division from left to right: \(72 \div 12 = 6\), and then \(6 \div 3 = 2\). 2. For Expression B, evaluate the parentheses first: \(12 \div 3 = 4\), and then \(72 \div 4 = 18\). 3. The values differ because the parentheses in Expression B require the second division to be completed first, while Expression A is evaluated from left to right.

Expression A has a value of \(2\). Expression B has a value of \(18\). The parentheses change which division is performed first.

- Evaluate exponents before multiplication. - Rewrite each power as a product of equal factors. - When there are two powers, evaluate each one separately before multiplying.

1. a) \(10^3 = 10 \times 10 \times 10 = 1000\), so \(7 \times 1000 = 7000\). 2. b) \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\), so \(4 \times 32 = 128\). 3. c) \(2^3 = 8\) and \(5^2 = 25\), so \(8 \times 25 = 200\).

a) \(7 \times (10 \times 10 \times 10) = 7000\) b) \(4 \times (2 \times 2 \times 2 \times 2 \times 2) = 128\) c) \((2 \times 2 \times 2) \times (5 \times 5) = 200\)

- Evaluate exponents before the other operations. - For a two-digit result, add its two digits. - Match each digit sum to a letter in the table.

1. Evaluate the expressions: a) \(64 - 45 = 19\); b) \(81 - 74 = 7\); c) \(25 - 10 = 15\); d) \(8 + 1 = 9\); e) \(100 \div 50 = 2\). 2. Add the digits: a) \(1 + 9 = 10\); b) \(7\); c) \(1 + 5 = 6\); d) \(9\); e) \(2\). 3. The table gives the letters S, P, O, R, T.

SPORT

- With equal positive exponents, compare the positive bases. - With the same base greater than \(1\), compare the exponents. - In part c, rewrite both powers as products and identify a common positive factor.

1. a) The exponents are equal and the positive base \(18\) is greater than \(17\), so \(17^2 < 18^2\). 2. b) The bases are equal and greater than \(1\). Since \(6 > 5\), \(10^5 < 10^6\). 3. c) Rewrite the expressions as products: \(5^2 \times 2 = 5 \times 5 \times 2\) and \(5 \times 2^2 = 5 \times 2 \times 2\). Both contain the positive factor \(5 \times 2\). Comparing the remaining factors, \(5 > 2\), so \(5^2 \times 2 > 5 \times 2^2\).

a) \(<\) b) \(<\) c) \(>\)

- In part a, use the order of operations and evaluate the parentheses first. - Rewrite \(20^2\) as a product and compare it with the product on the right. - When different amounts are subtracted from the same number, which expression leaves the greater result?

1. a) The left side is \(8^2 = 8 \times 8\). The right side is \(4^2 + 4^2 = 4 \times 4 + 4 \times 4 = 4 \times 8\). Since \(8 \times 8 > 4 \times 8\), the left side is greater. 2. b) \(20^2 = 20 \times 20\). Also, \(40 \times 10 = 2 \times 20 \times 10 = 20 \times 20\), so the expressions are equal. 3. c) Both expressions begin with \(10^4\). The left side subtracts \(10^2\), while the right side subtracts the larger amount \(10^3\). Therefore, the left side is greater.

a) \(>\) b) \(=\) c) \(>\)

- How many patterns are possible with one switch? With two switches? - Each additional switch doubles the number of patterns. - Compare \(2^8\) and \(2^9\) with \(365\).

1. Each switch has \(2\) possible states. With \(n\) switches, the number of patterns is \(2^n\). 2. a) \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\). 3. b) \(2^{10} = 1024\). 4. c) Find the smallest power of \(2\) that is at least \(365\). Since \(2^8 = 256\) is too small and \(2^9 = 512\) is large enough, at least \(9\) switches are needed.

a) \(2^5 = 32\) patterns. b) \(2^{10} = 1024\) patterns. c) At least \(9\) switches.

- Evaluate powers before addition or subtraction. - Work from the innermost grouping symbols outward. - Pay close attention when subtracting a negative number.

1. Evaluate the powers: \(4^2=16\), \(5^2=25\), and \(6^2=36\). 2. Evaluate the inner parentheses: \(25-36=-11\). 3. Evaluate the brackets: \(12-16-(-11)=-4+11=7\). 4. Subtract: \(40-7=33\).

\(33\)

- Consider which operation would increase the value most if completed before multiplication. - Test every placement that changes the usual order of operations. - Compare the resulting values.

1. For the greatest value, group the first sum: \((14 + 6) \times 7 - 5 = 20 \times 7 - 5 = 135\). 2. For the least value, group the final difference: \(14 + 6 \times (7 - 5) = 14 + 6 \times 2 = 26\). 3. Checking every distinct placement of one pair of parentheses confirms that no value is greater than \(135\) or less than \(26\).

Greatest value: \((14 + 6) \times 7 - 5 = 135\) Least value: \(14 + 6 \times (7 - 5) = 26\)

- Parentheses can make an addition or subtraction occur before multiplication. - Consider how grouping the terms after the subtraction sign changes the amount subtracted. - Try to make the subtracted product equal to \(40\) for the least value.

1. Without added parentheses, the value is \(40 - 4 \times 8 + 2 = 40 - 32 + 2 = 10\). 2. For the greatest value, group the first difference: \((40 - 4) \times 8 + 2 = 36 \times 8 + 2 = 290\). 3. For the least value, group the final sum: \(40 - 4 \times (8 + 2) = 40 - 4 \times 10 = 0\). 4. Checking every distinct placement of one pair of parentheses confirms that \(290\) is greatest and \(0\) is least.

Greatest value: \((40 - 4) \times 8 + 2 = 290\) Least value: \(40 - 4 \times (8 + 2) = 0\)

- Complete operations inside parentheses first. - Apply multiplication before subtraction. - When multiplication and division occur in sequence, evaluate from left to right. - Record intermediate values to keep the work organized.

1. \(14 \times 5 = 70\) and \(6 \times 7 = 42\), so \(70 - 42 = 28\). 2. \(23 \times 3 = 69\) and \(8 \times 4 = 32\), so \(69 - 32 = 37\). 3. Evaluate from left to right: \(28 \times 3 \div 6 = 84 \div 6 = 14\). 4. Evaluate from left to right: \(15 \times 4 \div 5 = 60 \div 5 = 12\). 5. \(12 \times 8 = 96\) and \(9 \times 9 = 81\), so \(96 - 81 = 15\).

1) \(28\) 2) \(37\) 3) \(14\) 4) \(12\) 5) \(15\)

- Evaluate the left side and the right side separately. - Apply the order of operations to each expression. - Compare the two final values.

1. The left side is \((15 \times 5) - 30 = 75 - 30 = 45\). The right side is \(5 \times 8 = 40\), so \(45 > 40\). 2. The left side is \(54 \div 6 \times 3 = 9 \times 3 = 27\). The right side is \((12 \times 2) + 5 = 24 + 5 = 29\), so \(27 < 29\). 3. The left side is \((19 \times 2) - 15 = 38 - 15 = 23\). The right side is \(3 \times 8 - 1 = 24 - 1 = 23\), so \(23 = 23\). 4. The left side is \(35 \div 5 \times 8 = 7 \times 8 = 56\). The right side is \((6 \times 8) + 2 = 48 + 2 = 50\), so \(56 > 50\).

1) \(>\) 2) \(<\) 3) \(=\) 4) \(>\)

- What rule applies when an expression contains parentheses? - How are addition and subtraction evaluated when there are no grouping symbols? - Track what happens to the \(10\) in each expression.

1. For Expression A, evaluate the parentheses first: \(30 + 10 = 40\). Then \(80 - 40 = 40\). 2. For Expression B, addition and subtraction have equal priority, so evaluate from left to right: \(80 - 30 = 50\), and then \(50 + 10 = 60\). 3. In Expression A, the entire sum \(30 + 10\) is subtracted from \(80\). In Expression B, \(30\) is subtracted and then \(10\) is added.

a) \(A = 40\); \(B = 60\) b) In Expression A, the parentheses make the sum \(30 + 10\) occur first, and that whole sum is subtracted. In Expression B, addition and subtraction are evaluated from left to right.

- Think about multiplying by a positive number less than \(1\). - A fraction such as one half can help you reason about what happens below \(1\). - Compare each starting value with \(1\).

1. For a positive number between \(0\) and \(1\), squaring makes the value smaller. For a number greater than \(1\), squaring makes the value larger. 2. For a), \(0.8<1\), and \(0.8^2=0.8\times0.8=0.64<0.8\). 3. For b), \(1.2>1\), and \(1.2^2=1.2\times1.2=1.44>1.2\). 4. For c), \(0.1<1\), and \(0.1^2=0.1\times0.1=0.01<0.1\). 5. For d), \(2.5>1\), and \(2.5^2=2.5\times2.5=6.25>2.5\).

a) Less; \(0.64\) b) Greater; \(1.44\) c) Less; \(0.01\) d) Greater; \(6.25\)

- Check each filled result from the top of the tree downward. - Recalculate the two-digit multiplication carefully. - After finding the first error, continue with the corrected value.

1. The addition is correct: \(15 + 18 = 33\). 2. The first error is the multiplication. The tree shows \(4 \times 33 = 122\), but \(4 \times 33 = 132\). 3. Using the correct intermediate value, \(250 - 132 = 118\).

a) The first error is the multiplication: \(4 \times 33 = 132\), not \(122\). b) \(118\)

- Read each tree from its branches toward the final operation. - Values joined directly by an operation form a subexpression. - Use parentheses to preserve the order shown in the tree.

1. For a), first add the values in the right branch: \(-25+(-15)=-40\). Then subtract this sum from \(12\). The expression is \(12-(-25+(-15))\), and its value is \(12-(-40)=52\). 2. For b), evaluate the two branches: \(-8-14=-22\) and \(-11+6=-5\). Then add the results. The expression is \((-8-14)+(-11+6)\), and its value is \(-22+(-5)=-27\).

a) \(12-(-25+(-15))=52\) b) \((-8-14)+(-11+6)=-27\)

- Identify the operation performed in each branch before the final operation. - Parentheses show that those branch operations occur before multiplication. - Without parentheses, apply multiplication before addition or subtraction.

1. The left branch represents \(15+(-3)\), and the right branch represents \(4-(-7)\). The final operation multiplies these results, so the expression is \((15+(-3))\times(4-(-7))\). 2. The intermediate results are \(15+(-3)=12\) and \(4-(-7)=11\). The final result is \(12\times11=132\). 3. The parentheses require the addition and subtraction to occur before multiplication. Without them, the expression in standard notation is \(15-3\times4+7\), which equals \(15-12+7=10\).

a) \((15+(-3))\times(4-(-7))\) b) Intermediate results: \(12\) and \(11\); final result: \(132\) c) The parentheses preserve the tree’s operation order. Without them, the value is \(10\).

- Compare the order shown by the tree with the order in Lara’s expression. - Which two operations occur before the final division? - Use parentheses to make subtraction occur before division.

1. Lara omitted the parentheses that show both subtractions must be completed before the division. 2. The correct expression is \((48 - 12) \div (12 - 10)\). 3. Its value is \(36 \div 2 = 18\). 4. Lara’s expression is evaluated using the order of operations: \(48 - 12 \div 12 - 10 = 48 - 1 - 10 = 37\).

a) Lara omitted necessary parentheses. b) \((48 - 12) \div (12 - 10)\) c) The correct expression equals \(18\); Lara’s expression equals \(37\).

- Division is performed before subtraction when there are no grouping symbols. - Identify the first operation shown in each tree. - Evaluate each tree from its branches toward the final operation.

1. Division is performed before subtraction, so \(40\div5\) must be evaluated first. Tree A represents \(180-(40\div5)\), so Tree A is correct. Tree B represents \((180-40)\div5\). 2. Tree A: \(40\div5=8\), then \(180-8=172\). 3. Tree B: \(180-40=140\), then \(140\div5=28\).

a) Tree A b) Tree A: \(172\); Tree B: \(28\)

- Evaluate the calculation tree from the innermost branch toward the final operation.

1. Evaluate the product in the innermost branch: \(2\times9=18\). 2. Evaluate the quotient in the right branch: \(18\div3=6\). 3. Evaluate the final quotient: \(144\div6=24\). The expression is \(144\div((2\times9)\div3)\).

\(144\div((2\times9)\div3)=24\)

- Read the tree from the innermost branches toward the final operation. - Each branch operation may require parentheses in the written expression. - Use decimal points when writing the values.

1. The expression is \(1.5\times((8.2+1.8)\div5)\). 2. Evaluate the sum: \(8.2+1.8=10\). 3. Divide: \(10\div5=2\). 4. Multiply: \(1.5\times2=3\).

\(1.5\times((8.2+1.8)\div5)=3\)

- Evaluate the two branches independently. - Then add the two branch results. - Pay attention to the sign rules for multiplication and division.

1. Evaluate the left branch: \(6\times(-7)=-42\). 2. Evaluate the right branch: \(80\div(-16)=-5\). 3. Add the branch values: \(-42+(-5)=-47\). The expression is \(6\times(-7)+80\div(-16)\).

\(6\times(-7)+80\div(-16)=-47\)

- Recalculate every operation in the tree. - Check whether each box matches the operation above it. - Continue from the first error using the corrected value.

1. The left branch is correct: \(60 \times 5 = 300\). 2. The right branch is incorrect: \(480 \div 6 = 80\), not \(70\). The error occurred in division. 3. The correct final result is \(300 - 80 = 220\).

a) The error occurred in \(480 \div 6\). b) \(220\)

- Begin with the innermost operation in the right branch. - Use parentheses so the sum \(-8+3\) is evaluated before multiplication. - Add \(-15\) only after evaluating the right branch.

1. Evaluate the inner sum: \(-8+3=-5\). 2. Multiply: \((-5)\times(-4)=20\). 3. Add the left branch value: \(-15+20=5\). The expression is \(-15+((-8+3)\times(-4))\).

\(-15+((-8+3)\times(-4))=5\)

- Evaluate independent branches separately. - In the right branch, complete the multiplication before the subtraction. - Add the two branch results last.

1. Evaluate the left branch: \(12\div(-4)=-3\). 2. In the right branch, multiply first: \(3\times2=6\). 3. Complete the right branch: \(15-6=9\). 4. Add the branch values: \(-3+9=6\). The expression is \((12\div(-4))+(15-(3\times2))\).

\(6\)

- Buying another item should decrease, not increase, the change. - Check the operation at the final branch of the tree. - Subtract both purchase amounts from \(\$20\).

1. The tree first finds \(20 - 12 = 8\). It then adds the notebook cost, \(5 \times 1 = 5\), instead of subtracting it. 2. Lina’s tree gives \(8 + 5 = 13\), or \(\$13\). 3. Both costs must be subtracted: \(20 - 12 - 5 = 3\). Lina should receive \(\$3\).

a) Lina added the notebook cost instead of subtracting it. b) \(\$13\) c) \(\$3\)

- Write the difference between each pair of consecutive terms. - Look for a pattern in those differences.

1. The consecutive differences are \(16 - 9 = 7\), \(25 - 16 = 9\), \(36 - 25 = 11\), and \(49 - 36 = 13\). 2. The differences increase by \(2\), so the next difference is \(15\). 3. \(49 + 15 = 64\), and \(64 = 8^2\).

The differences are \(7, 9, 11, 13\), and the next perfect square is \(64\).

- Evaluate both endpoint values first. - Do not count either endpoint.

1. \(19^2 = 361\) and \(20^2 = 400\). 2. The number of whole numbers strictly between the endpoints is \(400 - 361 - 1 = 38\).

\(38\) whole numbers.

- Calculate the correct square independently. - Then subtract the two results to find the error.

1. \(32^2 = 32 \times 32 = 1024\). 2. The difference is \(1024 - 924 = 100\). 3. The incorrect result is \(100\) too small.

The correct value is \(1024\), and \(924\) is \(100\) too small.

- Test the claim with two odd whole numbers. - Then use the fact that the product of two odd numbers is odd.

1. For example, \(9^2 = 81\) and \(13^2 = 169\), and both results are odd. 2. Squaring an odd whole number means multiplying that odd number by itself. The product of two odd whole numbers is always odd, so the square of every odd whole number is odd.

The statement is true. For example, \(9^2 = 81\) and \(13^2 = 169\), and the product of two odd numbers is always odd.

- Compare two consecutive perfect squares. - Find their difference before subtracting it from \(20^2\).

1. \(20^2 = 400\). 2. The difference between \(20^2\) and \(19^2\) is \(20 + 19 = 39\). 3. Therefore, \(19^2 = 400 - 39 = 361\).

Subtract \(39\); \(19^2 = 361\).

- Look for two equal factors of \(1156\). - Check that your square arrangement uses all the dots.

1. Find a whole number \(n\) such that \(n^2 = 1156\). 2. Since \(34^2 = 1156\), there are \(34\) dots in each row.

\(34\) dots.

- Find the total number of items in each square array. - Then subtract the new total from the original total.

1. The original array has \(20^2 = 400\) items. 2. The new array has \(18^2 = 324\) items. 3. The decrease is \(400 - 324 = 76\).

\(76\) fewer items.

- Evaluate the two powers separately. - Compare their sum with familiar perfect squares.

1. \(9^2 = 81\) and \(12^2 = 144\). 2. \(81 + 144 = 225\). 3. Since \(15^2 = 225\), the result is a perfect square.

\(225 = 15^2\); yes, it is a perfect square.

- Evaluate the two perfect squares separately. - Check both values before subtracting.

1. \(40^2 = 1600\). 2. \(38^2 = 1444\). 3. \(1600 - 1444 = 156\).

\(40^2 - 38^2 = 156\).

- Isolate the unknown perfect-square value first. - Then identify the whole-number base of that perfect square.

1. Subtract \(25\): \(n^2 = 169 - 25 = 144\). 2. Since \(12^2 = 144\), \(n = 12\).

\(n = 12\)

- First find the value of the missing perfect square. - Then identify the whole-number base whose square has that value.

1. \(15^2 = 225\). 2. The missing perfect square is \(225 + 31 = 256\). 3. Since \(256 = 16^2\), the missing base is \(16\).

\(16\)

- Evaluate the known power first. - Check whether the missing addend is a perfect square.

1. \(13^2 = 169\). 2. The missing number is \(425 - 169 = 256\). 3. Since \(256 = 16^2\), the missing perfect square is \(256\) and its base is \(16\).

\(256 = 16^2\).

- First find the value of the unknown perfect square. - Then identify its whole-number base and check it in the original equation.

1. \(25^2 = 625\). 2. The equation becomes \(625 - n^2 = 264\), so \(n^2 = 625 - 264 = 361\). 3. Since \(19^2 = 361\), \(n = 19\).

\(n = 19\)

- Use nearby perfect squares to narrow the possible base. - Check whether more than one perfect square fits the interval.

1. \(22^2 = 484 < 500\). 2. \(23^2 = 529\), which lies between \(500\) and \(550\). 3. \(24^2 = 576 > 550\), so \(529\) is the only perfect square in the interval.

\(529 = 23^2\).

- First find all perfect squares in the interval. - Then apply the condition about the base.

1. Nearby perfect squares are \(26^2 = 676\), \(27^2 = 729\), \(28^2 = 784\), and \(29^2 = 841\). 2. The perfect squares in the interval are \(729\) and \(784\). Their bases are \(27\) and \(28\), respectively. 3. Since \(28\) is even, the required perfect square is \(784\).

\(784 = 28^2\).

- List the perfect squares in the given interval. - Then check the ones digit of each candidate.

1. The perfect squares in the interval are \(11^2 = 121\), \(12^2 = 144\), \(13^2 = 169\), and \(14^2 = 196\). 2. Only \(121\) has a ones digit of \(1\).

\(121 = 11^2\).

- First make a complete list of the two-digit perfect squares. - Check the digit-sum condition for every candidate.

1. The two-digit perfect squares are \(16, 25, 36, 49, 64, 81\). 2. Their digit sums are \(7, 7, 9, 13, 10, 9\), respectively. 3. Therefore, the numbers that satisfy the condition are \(36 = 6^2\) and \(81 = 9^2\).

\(36\) and \(81\).

- First find all perfect squares in the interval. - Then check whether each base is odd or even.

1. The perfect squares in the interval are \(15^2 = 225\), \(16^2 = 256\), and \(17^2 = 289\). 2. The odd bases are \(15\) and \(17\). 3. Therefore, the required perfect squares are \(225\) and \(289\).

\(225 = 15^2\) and \(289 = 17^2\).

- Find where perfect squares change from three digits to four digits. - Compare the two consecutive perfect squares at that boundary.

1. \(31^2 = 961\), which is a three-digit number. 2. \(32^2 = 1024\), which is a four-digit number. 3. Therefore, \(961\) is the greatest three-digit perfect square.

\(961 = 31^2\).

- Find the first and last bases whose squares are in the interval. - Count the whole numbers in that inclusive range of bases.

1. The first perfect square in the interval is \(10^2 = 100\). 2. The last is \(22^2 = 484\), because \(23^2 = 529 > 500\). 3. The whole-number bases from \(10\) through \(22\) give \(22 - 10 + 1 = 13\) perfect squares.

\(13\) perfect squares.

- First identify the least and greatest possible bases. - Then check every whole-number base in between.

1. The smallest possible base is \(13\), because \(12^2 = 144 < 150\). 2. The greatest possible base is \(18\), because \(19^2 = 361 > 350\). 3. The perfect squares are \(13^2 = 169\), \(14^2 = 196\), \(15^2 = 225\), \(16^2 = 256\), \(17^2 = 289\), and \(18^2 = 324\).

\(169, 196, 225, 256, 289\), and \(324\).

- Find the perfect square immediately below and immediately above \(250\). - Calculate the two distances separately.

1. The neighboring perfect squares are \(15^2 = 225\) and \(16^2 = 256\). 2. Therefore, \(225 < 250 < 256\). 3. The distances are \(250 - 225 = 25\) and \(256 - 250 = 6\).

\(225 < 250 < 256\); the distances are \(25\) and \(6\).

- Place \(72\) between two consecutive perfect squares. - Compare it with the greater of those two squares.

1. The array has \(72\) dots, and \(8^2 = 64 < 72 < 81 = 9^2\). 2. The next greater perfect square is \(81\). 3. \(81 - 72 = 9\), so \(9\) dots must be added.

Add \(9\) dots to make \(81 = 9^2\) dots.

- Find the current total and the target total. - Subtract to compare the two arrays.

1. The existing grid has \(9 \times 13 = 117\) small squares. 2. The completed square will have \(13^2 = 169\) small squares. 3. \(169 - 117 = 52\), so \(52\) small squares are missing.

\(52\) small squares.

- Find the perfect square immediately below and immediately above \(86\). - Compare the two required changes.

1. The neighboring perfect squares are \(9^2 = 81\) and \(10^2 = 100\). 2. To reach \(81\), remove \(86 - 81 = 5\) dots. 3. To reach \(100\), add \(100 - 86 = 14\) dots. 4. Removing dots is more efficient because \(5 < 14\).

Remove \(5\) dots to make \(81 = 9^2\), or add \(14\) dots to make \(100 = 10^2\). Removing dots requires the smaller change.

- Multiplication usually changes a value more than addition or subtraction. - Think about how parentheses can make the factor multiplied by \(5\) larger. - To make the result smaller, consider grouping terms so that a larger quantity is subtracted.

1. Without added parentheses, the value is \(10+5\times1.2-0.2+4=19.8\). 2. For the greatest value, use \(10+5\times(1.2-0.2+4)\). The value is \(10+5\times5=35\). 3. For the least value, use \(10+5\times1.2-(0.2+4)\). The value is \(10+6-4.2=11.8\). 4. In part a), the parentheses make \(5\) multiply the entire value \(1.2-0.2+4=5\), rather than only \(1.2\).

a) \(10+5\times(1.2-0.2+4)=35\) b) \(10+5\times1.2-(0.2+4)=11.8\) c) The parentheses make \(5\) multiply a larger quantity.

- Without added parentheses, multiply before adding or subtracting. - Test each possible meaningful placement of one pair of parentheses. - Remember that among negative numbers, a value closer to zero is greater.

1. Without added parentheses, multiply first: \(5\times0.4=2\). Then \(-2.5+2-1.2=-1.7\). 2. The greatest value is obtained with \((-2.5+5)\times0.4-1.2\). This equals \(2.5\times0.4-1.2=1-1.2=-0.2\). 3. The least value is obtained with \(-2.5+5\times(0.4-1.2)\). This equals \(-2.5+5\times(-0.8)=-6.5\).

a) \(-1.7\) b) \((-2.5+5)\times0.4-1.2=-0.2\) c) \(-2.5+5\times(0.4-1.2)=-6.5\)

- Compare each grouped expression with the original value. - Parentheses force the enclosed operations to be completed first. - In part c, make the quantity subtracted from \(500\) equal \(200\).

1. a) Evaluate from left to right: \(500 - 150 + 100 - 50 = 400\). 2. b) With the stated parentheses, \(500 - (150 + 100) - 50 = 500 - 250 - 50 = 200\). Since \(200 < 400\), the claim is true. 3. c) \(500 - (150 + 100 - 50) = 500 - 200 = 300\).

a) \(400\) b) The claim is true because the new value is \(200\). c) \(500 - (150 + 100 - 50) = 300\)

- First evaluate the expression without parentheses. - Grouping a sum after a minus sign causes the entire sum to be subtracted. - Part d requires two separate pairs of parentheses.

1. a) \(80 - (30 + 20) - 10 - 5 = 15\). 2. b) \(80 - (30 + 20 - 10) - 5 = 35\). 3. c) \(80 - 30 + 20 - (10 - 5) = 65\). 4. d) \(80 - (30 + 20) - (10 - 5) = 25\).

a) \(80 - (30 + 20) - 10 - 5 = 15\) b) \(80 - (30 + 20 - 10) - 5 = 35\) c) \(80 - 30 + 20 - (10 - 5) = 65\) d) \(80 - (30 + 20) - (10 - 5) = 25\)

- Parentheses can change the normal order of operations. - Place a sum in parentheses when it should be multiplied as one quantity. - Check whether part d already has the required value.

1. a) \(15 \times (4 + 2) \times 3 = 15 \times 6 \times 3 = 270\). 2. b) \(15 \times (4 + 2 \times 3) = 15 \times 10 = 150\). 3. c) \((15 \times 4 + 2) \times 3 = 62 \times 3 = 186\). 4. d) No parentheses are needed: \(15 \times 4 + 2 \times 3 = 60 + 6 = 66\).

a) \(15 \times (4 + 2) \times 3 = 270\) b) \(15 \times (4 + 2 \times 3) = 150\) c) \((15 \times 4 + 2) \times 3 = 186\) d) \(15 \times 4 + 2 \times 3 = 66\)

- Pay close attention to the entire divisor after a division symbol. - Evaluate the innermost parentheses first. - The order of operations still applies inside parentheses.

1. a) \(72 \div (6 + 2) \times 3 = 72 \div 8 \times 3 = 27\). 2. b) \(72 \div (6 + 2 \times 3) = 72 \div 12 = 6\). 3. c) \((72 \div 6 + 2) \times 3 = 14 \times 3 = 42\). 4. d) \(72 \div ((6 + 2) \times 3) = 72 \div 24 = 3\).

a) \(72 \div (6 + 2) \times 3 = 27\) b) \(72 \div (6 + 2 \times 3) = 6\) c) \((72 \div 6 + 2) \times 3 = 42\) d) \(72 \div ((6 + 2) \times 3) = 3\)

- Pay close attention to each subtraction sign before a grouped expression. - Compare the grouped calculation with what would happen if you evaluated from left to right without that grouping. - Test the parentheses and the square brackets separately.

1. One reasonable estimate is \(12{,}000 - [4000 - (1000 + 1000)] = 12{,}000 - 2000 = 10{,}000\). 2. Evaluate the parentheses: \(1150 + 650 = 1800\). 3. Evaluate the square brackets: \(4320 - 1800 = 2520\). 4. Subtract: \(12{,}450 - 2520 = 9930\). 5. Removing the parentheses would change the expression inside the brackets to \(4320 - 1150 + 650\), which equals \(3820\), not \(2520\). Removing the square brackets would also change which quantity is subtracted from \(12{,}450\). Therefore, both pairs of grouping symbols are necessary.

Estimate: \(\approx 10{,}000\) Exact value: \(9930\) Neither pair of grouping symbols can be removed.

- Recall that multiplication and division are usually completed before addition and subtraction. - Parentheses are evaluated first. - Choose numbers so that removing the parentheses changes which operation is performed first. - Try combining addition with multiplication or division.

1. One expression with a value of \(15\) is \((2 + 3) \times 3\). With the parentheses, \((2 + 3) \times 3 = 5 \times 3 = 15\). Without them, \(2 + 3 \times 3 = 2 + 9 = 11\), so the value changes. 2. One expression with a value of \(25\) is \((20 + 30) \div 2\). With the parentheses, \((20 + 30) \div 2 = 50 \div 2 = 25\). Without them, \(20 + 30 \div 2 = 20 + 15 = 35\), so the value changes. 3. Other answers are possible if they meet all the stated conditions.

Possible answers: For \(15\): \((2 + 3) \times 3 = 15\), while \(2 + 3 \times 3 = 11\). For \(25\): \((20 + 30) \div 2 = 25\), while \(20 + 30 \div 2 = 35\).

- Test each pair of grouping symbols separately. - Recall that operations of equal priority are evaluated from left to right. - Grouping after subtraction or division is often important because it can change the quantity being subtracted or used as the divisor.

1. a) The parentheses around \(25 - 5\) and the square brackets can be removed. The parentheses around \(10 - 2\) must remain because the entire difference is subtracted. Thus, \(25 - 5 - (10 - 2) + 5 = 20 - 8 + 5 = 17\). 2. b) The parentheses around \(8 \times 5\) and the square brackets can be removed. The parentheses around \(2 \times 3\) must remain because \(120\) is divided by the entire product. Thus, \(8 \times 5 + 120 \div (2 \times 3) = 40 + 120 \div 6 = 60\). 3. c) The square brackets can be removed, but the parentheses around \(14 + 16\) must remain so the addition occurs before multiplication. Thus, \((14 + 16) \times 2 \div 4 = 30 \times 2 \div 4 = 15\).

a) \(25 - 5 - (10 - 2) + 5 = 17\) b) \(8 \times 5 + 120 \div (2 \times 3) = 60\) c) \((14 + 16) \times 2 \div 4 = 15\)

- Work systematically so that you do not miss a possible grouping. - Consider which consecutive parts of the expression can be evaluated first. - Include nested grouping, and compare the resulting values.

1. Preserving the original operation order gives \(2 \times 10 - 4 + 6 = 22\). 2. Grouping \(10 - 4\) gives \(2 \times (10 - 4) + 6 = 18\). 3. Grouping \(4 + 6\) gives \(2 \times 10 - (4 + 6) = 10\). 4. Grouping \(10 - 4 + 6\) gives \(2 \times ((10 - 4) + 6) = 24\). 5. Nested grouping gives \(2 \times (10 - (4 + 6)) = 0\). 6. These are all five full groupings of the numbers and operations in their original order, so no other result is possible.

The different whole-number results are \(0\), \(10\), \(18\), \(22\), and \(24\).

- More than one pair of parentheses may be needed. - Work backward from each target value to identify useful factors. - Test possible groupings systematically and evaluate each result.

1. a) Group the middle sum: \(3 \times (16 + 14) \times 2 = 3 \times 30 \times 2 = 180\). 2. b) Group both differences: \((60 - 20) \times (12 - 7) = 40 \times 5 = 200\). 3. c) Group the middle difference: \(15 \times (10 - 8) + 4 = 15 \times 2 + 4 = 34\).

a) \(3 \times (16 + 14) \times 2 = 180\) b) \((60 - 20) \times (12 - 7) = 200\) c) \(15 \times (10 - 8) + 4 = 34\)

- In which direction are consecutive divisions evaluated? - Can division be distributed over a sum in the divisor? - Evaluate the parentheses first in part b.

1. a) The calculation combines \(20 \div 5\) first. Consecutive divisions must be evaluated from left to right: \(200 \div 20 = 10\), then \(10 \div 5 = 2\). 2. b) Division cannot be distributed over a sum in the divisor. Evaluate the parentheses first: \(100 \div (10 + 10) = 100 \div 20 = 5\).

a) The expression was evaluated from right to left. The correct result is \(2\). b) Division was incorrectly distributed over the sum in the divisor. The correct result is \(5\).

- Think of each answer as one branch of a decision tree. - How many answer paths are possible after one, two, and three questions? - Compare the number of answer paths with the \(100\) possible secret numbers. - Find the first power of \(2\) that is at least \(100\).

1. Each yes-or-no question creates two possible answer paths. Therefore, \(n\) questions can distinguish at most \(2^n\) possibilities. 2. a) \(2^6 = 64\), so \(6\) questions can distinguish at most \(64\) possibilities. 3. b) There are \(100\) possible secret numbers, and \(64 < 100\). Therefore, \(6\) questions cannot distinguish all possible numbers. 4. c) Find the smallest \(n\) for which \(2^n \geq 100\). Since \(2^6 = 64\) and \(2^7 = 128\), the least possible value is \(n = 7\).

a) At most \(2^6 = 64\) possibilities. b) Since \(64 < 100\), \(6\) questions cannot distinguish all \(100\) possible numbers. c) At least \(7\) questions.

- Think about how increasing a divisor affects a quotient. - Consider how to make the final multiplication apply to a larger quantity. - Test each meaningful placement and compare the values.

1. For the greatest value, make the sum a factor: \((36 \div 3 + 3) \times 2 = (12 + 3) \times 2 = 30\). 2. For the least value, make the entire sum after the division sign the divisor: \(36 \div (3 + 3 \times 2) = 36 \div (3 + 6) = 4\). 3. Checking all distinct placements of one pair of parentheses confirms that \(30\) is greatest and \(4\) is least.

a) \((36 \div 3 + 3) \times 2 = 30\) b) \(36 \div (3 + 3 \times 2) = 4\)

- Use the order of operations. - Evaluate the expressions on the two sides of the plus sign separately. - Track whether each result has a unit. - Can quantities of different types be added?

1. Evaluate the first part: \(4\,\text{hr} \div 2 = 2\,\text{hr}\). Dividing a duration by a number gives another duration. 2. Evaluate multiplication and division in the second part from left to right: \(10 \times 5\,\text{min} = 50\,\text{min}\), and \(50\,\text{min} \div 50\,\text{min} = 1\). Dividing equal units gives a unitless number. 3. The expression becomes \(2\,\text{hr} + 1\). 4. A duration cannot be added to a unitless number, so the expression has no valid overall value.

The expression cannot be evaluated to one valid value. The first part is the duration \(2\,\text{hr}\), while the second part is the unitless number \(1\); these quantities cannot be added.

- Identify the sequence of last digits that repeats. - Count how many terms are in one complete cycle. - Divide the exponent by the cycle length and use the remainder.

1. The last digits begin \(2, 4, 8, 6, 2, 4, 8, 6\). 2. The cycle \(2, 4, 8, 6\) repeats every \(4\) exponents. 3. Divide the exponent by the cycle length: \(50 \div 4 = 12\) remainder \(2\). 4. A remainder of \(2\) means the last digit matches the second number in the cycle, which is \(4\).

The last digit of \(2^{50}\) is \(4\).

- Relate the difference between consecutive perfect squares to the sum of their bases. - Verify the two squares by subtracting.

1. The difference between the squares of two consecutive whole numbers equals the sum of those numbers. 2. Since \(25 + 26 = 51\), the bases are \(25\) and \(26\). 3. \(25^2 = 625\), \(26^2 = 676\), and \(676 - 625 = 51\).

\(625 = 25^2\) and \(676 = 26^2\).

- First narrow the possible whole-number bases. - Then use the condition about the final two digits.

1. The base must be between \(32\) and \(44\), because \(32^2 = 1024\), \(44^2 = 1936\), and \(45^2 = 2025\). 2. A perfect square that ends in two zeros is divisible by \(100\), so its whole-number base must be divisible by \(10\). The only multiple of \(10\) from \(32\) through \(44\) is \(40\). 3. \(40^2 = 1600\), which satisfies all the conditions.

\(1600 = 40^2\).