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

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

- Match each position in the sequence with the whole number being squared. - A complete answer needs both the missing numerical value and its exponent form. - Check that every exponent is \(2\).

1. The sequence lists the perfect squares from \(1^2\) through \(10^2\). 2. The first missing value is \(4^2=16\). 3. The second missing value is \(7^2=49\). 4. The third missing value is \(10^2=100\).

\(4^2=16\), \(7^2=49\), and \(10^2=100\).

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

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

- 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 \(15^2 = 225\).

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

- Do not simply calculate from left to right when operations differ. - Which operation has priority over addition?

1. Multiply first: \(3\times4=12\). 2. Add: \(8+12=20\).

\(20\). Multiplication comes before addition.

- What quantity is enclosed by the parentheses? - Use the entire result as the divisor.

1. Evaluate the grouped sum: \(2+4=6\). 2. Divide the whole numerator by that sum: \(24\div6=4\).

\(4\). The parentheses require the addition first.

- How many copies of the base does the exponent ask for? - Compare the repeated-factor expression with ordinary multiplication.

1. The power \(3^2\) means \(3\times3=9\). 2. The multiplication \(3\times2=6\), so the values are different.

\(3^2=9\) and \(3\times2=6\); they are not equal. An exponent counts equal factors.

- Think about how many equal factors the exponent counts. - What is a product containing only one factor?

1. An exponent of \(1\) means the base appears as one factor. 2. Therefore, \(12^1=12\).

\(12\). There is one factor of \(12\).

- Rewrite the power as repeated multiplication. - Count the zeros in the result.

1. \(10^3=10\times10\times10=1000\). 2. There are three zeros after the \(1\).

\(1000\); three zeros.

- Compare the priority of multiplication and division. - Which operation appears first as you read from left to right?

1. Multiplication and division have the same priority, so work from left to right. 2. \(20\div5=4\), then \(4\times2=8\). 3. Multiplying \(5\times2\) first would introduce parentheses that are not in the original expression.

\(8\). Multiplication and division have equal priority and are done from left to right.

- Follow the branches toward the final result. Which branch must have a value before the final calculation can happen? - How will that result be used by the other operation?

1. The inner branch multiplies \(2\times3=6\). 2. The final operation subtracts this product from \(12\): \(12-6=6\).

Multiply first: \(2\times3\). Final value: \(6\).

- 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: \(300 - 20 + 3 = 283\).

\(283\)

- 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 \(B=4\).

- 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. \(0.7^2=0.49\). 2. Since \(3^4=81\), \(\left(\frac{1}{3}\right)^4=\frac{1}{81}\). 3. \(0.2^3=0.008\). 4. \(1.1\times1.1=1.21\).

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. - Compare the three final values only after each expression is fully evaluated.

1. Expression A: \(\frac{1}{4}\times20-2+3=5-2+3=6\). 2. Expression B: \(\frac{1}{4}\times(20-2)+3=\frac{1}{4}\times18+3=4.5+3=7.5\). 3. Expression C: \(\frac{1}{4}\times20-(2+3)=5-5=0\). 4. Since \(0<6<7.5\), Expression C has the least value.

Expression A: \(6\) Expression B: \(7.5\) Expression C: \(0\) 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 difference inside the parentheses first. - A subtraction sign before a grouped difference changes the effect of both terms when the grouping is removed. - Compare each student’s result with a direct evaluation of the original expression.

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

Maya is correct. Lucas should have changed the second subtraction to addition when removing the parentheses. The correct value is \(10\).

- Evaluate each set of parentheses first. - Identify which subtraction is completed before the other one. - Compare the role of the final \(3\) in the two expressions.

1. Expression A: \(12-(5-3)=12-2=10\). 2. Expression B: \((12-5)-3=7-3=4\). 3. In Expression A, the subtraction sign applies to the complete difference \(5-3\). In Expression B, the first subtraction is completed before subtracting \(3\).

a) Expression A is \(10\), and Expression B is \(4\). b) The grouping symbols make different subtractions occur first.

- 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 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) \(<\)

- 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) \(>\)

- 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) \(>\)

- 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) \(25\)

- A square array has the same number of rows and columns. - Find a whole number that gives \(81\) when multiplied by itself. - Use the side length as the base of the square power.

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. \(80 \div 4 \times 2 = 40\).

- 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 \(10 \div 2 = 5\).

- Compare the original expression's operation order with the order used in Minh's calculation. - Ask whether the original expression contains parentheses that justify doing \(40-10\) first. - Check the final value by following standard order-of-operations conventions.

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. Minh effectively inserted parentheses to create \(100-(40-10)\), which is not the original expression.

Minh's claim is incorrect. \(100-40-10=50\).

- 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. \(10^5=10\times10\times10\times10\times10=100{,}000\). 2. \(1^{100}=\underbrace{1\times1\times\cdots\times1}_{100\text{ factors}}=1\). 3. \(13^2=13\times13=169\). 4. \(2^6=2\times2\times2\times2\times2\times2=64\).

a) \(10\times10\times10\times10\times10=100{,}000\) b) \(\underbrace{1\times1\times\cdots\times1}_{100\text{ factors}}=1\) c) \(13\times13=169\) d) \(2\times2\times2\times2\times2\times2=64\)

- 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 each power before applying another operation. - Keep the base and exponent together when finding a power. - Check each expression independently.

1. For a), \(2^6=64\), so \(64-45=19\). 2. For b), \(3^4=81\), so \(81-74=7\). 3. For c), \(5^2=25\), so \(25-10=15\). 4. For d), \(2^3=8\) and \(1^5=1\), so \(8+1=9\). 5. For e), \(10^2=100\), so \(100\div50=2\).

a) \(2^6=64\); result \(19\) b) \(3^4=81\); result \(7\) c) \(5^2=25\); result \(15\) d) \(2^3=8\) and \(1^5=1\); result \(9\) e) \(10^2=100\); result \(2\)

- 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 \(2^9 = 512\).

- Evaluate multiplication before addition or subtraction. - Use the multiplication properties of \(0\) and \(1\). - In part e, notice the factor outside the grouping symbols.

1. For a), \(0\times13+7=0+7=7\). 2. For b), \(25-0\times4=25-0=25\). 3. For c), \(6\times0+8\times1=0+8=8\). 4. For d), \(1\times9-9\times0=9-0=9\). 5. For e), any product with factor \(0\) is \(0\), so the value is \(0\).

a) \(7\) b) \(25\) c) \(8\) d) \(9\) e) \(0\)

- 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) \(0.000225\)

- Identify the multiplication branch first. - Add \(26\) to that intermediate value. - Use the tree to check the order of operations.

1. Multiply: \(9\times6=54\). 2. Add: \(54+26=80\). 3. The expression is \(9\times6+26\).

Intermediate value: \(54\) Expression: \(9\times6+26\) Final value: \(80\)

- Evaluate the product on the right branch first. - Subtract that product from \(60\). - The order of operations already matches the tree.

1. Multiply: \(8\times4=32\). 2. Subtract: \(60-32=28\). 3. The expression is \(60-8\times4\).

Intermediate value: \(32\) Expression: \(60-8\times4\) Final value: \(28\)

- The tree adds two products. - Evaluate each product before adding.

1. Evaluate the left product: \(8\times5=40\). 2. Evaluate the right product: \(6\times7=42\). 3. Add: \(40+42=82\). 4. The expression is \(8\times5+6\times7\).

Intermediate values: \(40\) and \(42\) Expression: \(8\times5+6\times7\) Final value: \(82\)

- 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 \(21^2 - 20^2 = 41\).

- First identify the whole-number base whose square equals each given number. - Then compare each number's distance from \(300\). - Give both exponent representations before making the distance comparison.

1. \(289=17^2\) and \(324=18^2\). 2. The distance from \(300\) to \(289\) is \(11\). 3. The distance from \(300\) to \(324\) is \(24\). 4. Since \(11<24\), \(289\) is closer to \(300\).

a) \(289=17^2\) and \(324=18^2\). b) \(289\) is closer to \(300\); the distances are \(11\) and \(24\).

- 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 \(144 \div 2 = 12\).

- Use the image to determine the equal row and column counts. - A square array can be represented by squaring its side length. - Include the base, exponent, and evaluated total in the answer.

1. The image shows \(8\) dots in each row and \(8\) dots in each column, so the side length is \(8\) dots. 2. The total is represented by \(8^2\), where the base is \(8\) and the exponent is \(2\). 3. \(8^2=64\), so the array has \(64\) dots.

The side length is \(8\) dots. The total is \(8^2=64\); the base is \(8\) and the exponent is \(2\).

- 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) \(49 = 7^2\)

- Compare each power with the next one in the list. - Continue the same change for one more step.

1. When the exponent drops by \(1\), the value is divided by \(9\). 2. Continuing from \(9^1=9\), divide by \(9\) once more to get \(9^0=1\).

\(9^0=1\). Each step down in the exponent divides the value by \(9\).

- Which number is directly attached to the exponent? - Are there any parentheses telling you to add first?

1. Theo changed the expression to \((4+2)^3\), which has different grouping. 2. The exponent applies only to \(2\): \(2^3=8\). 3. Then \(4+8=12\).

Theo grouped the wrong numbers before applying the exponent. The correct value is \(12\).

- Where does the exponent fit in the order of operations? - Try writing the value of each part of the expression before combining them.

1. Evaluate the power: \(3^2=9\). 2. Divide: \(18\div6=3\). 3. Add the two results: \(9+3=12\).

\(12\). Evaluate the power, divide, then add.

- 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 \(\left(\frac{1}{2}\right)^6=\frac{1}{64}\).

- Work from the innermost grouping symbols outward. - Evaluate multiplication and division before addition and subtraction. - When operations have the same priority, work from left to right.

1. For a), evaluate division and multiplication first: \(32\div4=8\) and \(12\times2=24\). Then \(8+24=32\). 2. For b), evaluate the brackets first: \(40-10=30\). Then \(150-30\times3=150-90=60\). 3. For c), evaluate multiplication and division first: \(0.2\times50=10\) and \(10\div2=5\). Then \(10+5=15\).

a) \(32\) b) \(60\) c) \(15\)

- Look for adjacent terms that can be grouped to change which operation happens first. - After evaluating the parentheses, perform division and multiplication from left to right. - Check the complete expression against the target value \(8\).

1. Put the parentheses around \(2+3\): \(20\div(2+3)\times2\). 2. Evaluate the parentheses: \(2+3=5\). 3. Divide and multiply from left to right: \(20\div5\times2=4\times2=8\).

\(20\div(2+3)\times2=8\)

- Evaluate exponents before subtraction. - Compare the square of a difference with the difference of two squares. - 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. \(0.2^2=0.04\), and \(0.04<0.2\). 2. \(\left(\frac{1}{2}\right)^3=\frac{1}{8}\) and \(\left(\frac{1}{2}\right)^2=\frac{1}{4}\), so \(\frac{1}{8}<\frac{1}{4}\). 3. \(0.1^2=0.01\), so the two values are equal. 4. \(\left(\frac{3}{2}\right)^2=\frac{9}{4}=2.25\), 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 which operation would happen first after each pair of symbols is removed. - Grouping around an already complete product may be unnecessary. - A subtraction sign before parentheses affects the entire grouped difference.

1. The parentheses around \(12+4\) cannot be removed. Without them, multiplication would happen before addition. 2. The brackets around the complete product can be removed: \((12+4)\times3-(10-2)\) has the same value. 3. The parentheses around \(10-2\) cannot be removed. The subtraction outside the parentheses applies to the entire difference.

1. No 2. Yes 3. 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) \(15-3\times(2.5+0.5)=6\)

- 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. - Evaluate the grouped difference first in Expression B. - When multiplication and division occur in sequence, work from left to right.

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

Expression A: \(3\) Expression B: \(30\) Expression B has the greater 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 first. - Complete multiplication and division before addition or subtraction. - Convert between fractions and decimals when it makes a calculation clearer.

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 \(0.25+1.5=1.75\). 3. For c), evaluate the parentheses: \(0.7-0.2=0.5\). Then \(\frac{2}{5}\div0.5=0.4\div0.5=0.8\).

a) \(0.1\) b) \(1.75\) c) \(0.8\), or \(\frac{4}{5}\)

- 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 \(10^4\) 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: \(48{,}500\)

- 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 \(450 - 50 = 400\).

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

- 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 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.

- 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: approximately \(4400\) 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: approximately \(5000\) Exact value: \(4950\) Both the parentheses and the square brackets can be removed.

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

1. Lucas 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.

Lucas 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.

- 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) \(44 - 4 \times (6 + 4) = 4\)

- 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) \(4 \times 4 - 4 \div 4 = 15\)

- 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 \(78\).

- Evaluate the exponent before the other operations. - Work from the innermost grouping symbols outward. - Complete multiplication before subtraction at each grouping level.

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, \(100-90=10\).

\(10\)

- 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) \(20 + 5 \times 3 = 35\)

- 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.

- In part a), consider squares of two consecutive positive numbers. - In part b), how does one additional factor of ten change a product? - In part c), write both expressions as products and compare their factors.

1. In a), the exponents match, but \(17<18\), so \(17^2<18^2\). 2. In b), the base is \(10>1\), so a higher exponent gives a greater value: \(10^5<10^6\). 3. In c), both products contain \(5\times2\). The remaining factors are \(5\) on the left and \(2\) on the right, so the left product is greater.

a) \(<\), because \(17<18\) and both numbers are squared. b) \(<\), because multiplying by another \(10\) increases the value. c) \(>\), because after matching \(5\times2\), the extra factor on the left is \(5\) instead of \(2\).

- In part a), do the addition inside the parentheses first. - Rewrite the square in part b) as repeated multiplication. - In part c), which expression subtracts more from the same starting value?

1. In a), the left side is \(8\times8\), while the right is \(4\times8\). Therefore, the left is greater. 2. In b), \(20^2=20\times20=400\) and \(40\times10=400\), so they are equal. 3. In c), both expressions start with \(10^4\), but the left subtracts the smaller number \(10^2\), leaving a greater result.

a) \(>\), because \(8\times8>4\times8\). b) \(=\), because both products equal \(400\). c) \(>\), because subtracting \(10^2\) leaves more than subtracting \(10^3\).

- 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. - Within one grouping level, evaluate addition and subtraction from left to right.

1. Evaluate the powers: \(4^2=16\), \(6^2=36\), and \(5^2=25\). 2. Evaluate the inner parentheses: \(36-25=11\). 3. Evaluate the brackets from left to right: \(28-16+11=12+11=23\). 4. Subtract: \(70-23=47\).

\(47\)

- Keep the numbers in the given order. - Use multiplication when you need a larger value. - Apply the order of operations when checking each expression. - More than one answer may be possible.

1. For a), one possible expression is \(2+4+6-8=4\). 2. For b), one possible expression is \(2\times4-6+8=10\). 3. For c), one possible expression is \(2+4\times6-8=18\). 4. For d), one possible expression is \(2+4+6+8=20\).

a) \(2+4+6-8=4\) b) \(2\times4-6+8=10\) c) \(2+4\times6-8=18\) d) \(2+4+6+8=20\)

- Complete the multiplication and division before the addition. - Convert the two time amounts to the same unit before adding. - One hour equals \(60\) minutes.

1. Evaluate multiplication and division first: \(4\,\text{hr}\div2=2\,\text{hr}\), and \(10\times5\,\text{min}=50\,\text{min}\). 2. Convert \(2\) hours to minutes: \(2\times60=120\,\text{min}\). 3. Add quantities with the same unit: \(120\,\text{min}+50\,\text{min}=170\,\text{min}\).

\(170\,\text{min}\)

- 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; \(6.25\)

- Begin with the two values joined by the innermost operation. - Follow the calculated results toward the final operation, checking each one. - After finding the first error, continue with the corrected value.

1. The addition is correct: \(15+18=33\). 2. The first error is in the multiplication. The tree gives \(4\times33=122\), but the correct product is \(132\). 3. Using the corrected value, \(250-132=118\).

a) The multiplication: \(4\times33=132\), not \(122\). b) \(118\)

- Work from the small branches toward the final result. - Numbers joined directly by an operation form a subexpression. - Use parentheses to preserve the order of operations.

1. In a), the right branch gives \(25+15=40\). The expression is \(52-(25+15)=12\). 2. In b), the branches give \(18-4=14\) and \(11+6=17\). The expression is \((18-4)+(11+6)=31\).

a) \(52-(25+15)=12\) b) \((18-4)+(11+6)=31\)

- Identify each branch operation before the final one. - How does writing the branches as a single expression change which parts belong together? - Compare the operation order in the tree with the usual order when parentheses are absent.

1. The two branches represent \(15+3\) and \(14-7\). The full expression is \((15+3)\times(14-7)\). 2. The intermediate values are \(18\) and \(7\), giving a final result of \(18\times7=126\). 3. Parentheses make the branch results come first. Without them, \(15+3\times14-7=15+42-7=50\).

a) \((15+3)\times(14-7)\) b) Intermediate results: \(18\) and \(7\); final result: \(126\) c) Parentheses preserve the tree's grouped operations. Without them, the value is \(50\).

- 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 \(37\).

- Which operation has priority when there are no parentheses? - Find the first operation in each tree. - Work from the branches toward the final result.

1. Division comes before subtraction, so the correct grouping is \(180-(40\div5)\), shown by Tree A. Tree B represents \((180-40)\div5\). 2. In Tree A, \(40\div5=8\), then \(180-8=172\). 3. In Tree B, \(180-40=140\), then \(140\div5=28\).

a) Tree A, because it divides \(40\) by \(5\) before subtracting from \(180\). b) Tree A: \(172\); Tree B: \(28\)

- Begin with the deepest branch of the tree. - Use parentheses to show that the inner product and quotient form the final divisor. - Evaluate one node at a time as you move toward the root.

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: \(1.5\times2=3\).

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

- 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) \(220\)

- 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: \(12\div4=3\). 2. In the right branch, multiply first: \(3\times2=6\). 3. Complete the right branch: \(15-6=9\). 4. Add the branch values: \(3+9=12\). The expression is \((12\div4)+(15-(3\times2))\).

\(12\)

- Buying more items should reduce the amount of change. - Look closely at the tree's last operation. - Find the total cost before calculating the actual change.

1. The tree calculates \(20-12=8\) and \(5\times1=5\). It then adds the notebook cost instead of subtracting it. 2. The tree gives \(8+5=13\) dollars. 3. The correct change is \(20-12-5=3\) dollars.

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

- Compare consecutive terms before predicting the next one. - After finding the numerical prediction, determine which whole number squared gives that value. - Part b) requires exponent notation, not just the number \(64\).

1. The consecutive differences are \(7\), \(9\), \(11\), and \(13\). 2. The differences increase by \(2\), so the next difference is \(15\). 3. The next term is \(49+15=64\). 4. Since \(64=8^2\), the base is \(8\) and the exponent is \(2\).

a) The differences are \(7,9,11,13\), so the next term is \(64\). b) \(64=8^2\); the base is \(8\).

- 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.

- The first strip runs along the full side of the larger square. - After the column is removed, how long is the bottom row? - Subtract the removed squares from the larger square's area.

1. The rightmost column contains \(20\) squares. The remaining bottom row contains \(19\) squares. 2. Together they contain \(20+19=39\) squares. 3. Since \(20^2=400\), the remaining area is \(400-39=361=19^2\).

a) Column: \(20\); row: \(19\); total removed: \(39\) b) \(19^2=361\)

- 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.

- Represent each square array using the common side length as the base. - Both powers should have exponent \(2\). - After evaluating the powers, compare the two totals.

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

\(20^2=400\) and \(18^2=324\), so the new array has \(76\) 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.

- 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 \(12^2 = 144\), \(n = 12\).

\(n = 12\)

- 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 \(16\).

\(256 = 16^2\).

- 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 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 \(81\).

- 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.

- Count the rows and columns of the displayed grid. - A square must have the same number of rows as columns. - Find the difference between the completed square and the displayed rectangle.

1. The rectangle has \(9\) rows and \(13\) columns, so it needs \(13-9=4\) more rows to make a square. 2. The completed square has \(13\times13\) small squares, so its total is \(13^2=169\). 3. The original rectangle has \(9\times13=117\) small squares. The number to add is \(169-117=52\).

Add \(4\) rows. Completed total: \(13^2=169\) small squares. Add \(52\) small squares.

- Find the largest perfect square below \(86\). - Find the smallest perfect square above \(86\). - Compare the distances to these two numbers.

1. The nearest perfect squares are \(9^2=81\) and \(10^2=100\). 2. Reaching \(81\) requires removing \(86-81=5\) dots. 3. Reaching \(100\) requires adding \(100-86=14\) dots. 4. Removing \(5\) dots is the smaller change.

Remove \(5\) dots to reach \(81=9^2\), or add \(14\) dots to reach \(100=10^2\). Removing dots changes fewer dots.

- Focus on placements that actually change which quantities are multiplied or subtracted together. - To maximize the result, consider how to make the quantity multiplied by \(5\) as large as possible. - To minimize the result, consider whether grouping can make a larger quantity be subtracted.

1. Without added parentheses, the value is \(10+5\times1.2-0.2+4=19.8\). 2. The value-changing legal placements are \((10+5)\times1.2-0.2+4=21.8\), \(10+5\times(1.2-0.2)+4=19\), \(10+5\times(1.2-0.2+4)=35\), and \(10+5\times1.2-(0.2+4)=11.8\). Other legal placements preserve the original order of operations and keep the value \(19.8\). 3. Therefore the greatest possible value is \(35\), from \(10+5\times(1.2-0.2+4)\), and the least possible value is \(11.8\), from \(10+5\times1.2-(0.2+4)\). 4. In the maximum placement, the factor \(5\) multiplies the entire quantity \(1.2-0.2+4=5\), 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.

- Compare each grouped expression with the original value. - Parentheses force the enclosed operations to be completed first. - For part c, work backward from the target and decide what the parenthesized group must accomplish.

1. Evaluate from left to right: \(500-150+100-50=400\). 2. With the stated parentheses, \(500-(150+100)-50=500-250-50=200\). Since \(200<400\), Efua's claim is true. 3. One valid placement is \(500-(150+100-50)=500-200=300\).

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

- 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: approximately \(10{,}000\) 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) \(15 \times (10 - 8) + 4 = 34\)

- To increase the value, look for a way to create two larger factors. - To decrease the value, make the product being subtracted as large as possible. - Check the distinct full groupings before deciding that the values are extreme.

1. To obtain the greatest value, make both factors as large as possible: \((20-2)\times(4+3)=18\times7=126\). 2. To obtain the least value, make the quantity subtracted from \(20\) as large as possible: \(20-2\times(4+3)=20-14=6\). 3. The other full groupings give values between \(6\) and \(126\), so these are the minimum and maximum.

Greatest value: \((20-2)\times(4+3)=126\) Least value: \(20-2\times(4+3)=6\)

- 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) \(36 \div (3 + 3 \times 2) = 4\)

- Keep the numbers in the given order. - Remember that multiplication and division are performed before addition and subtraction. - For \(31\), try making a factor close to \(3\). - Use grouping symbols when you need to change the usual order.

1. For a), one possible expression is \(10-5\times2+1=1\). 2. For b), one possible expression is \(10\div5+2\times1=4\). 3. For c), one possible expression is \(10-5+2\times1=7\). 4. For d), one possible expression is \(10+5-2\times1=13\). 5. For e), one possible expression is \(10\times(5-2)+1=31\).

a) \(10-5\times2+1=1\) b) \(10\div5+2\times1=4\) c) \(10-5+2\times1=7\) d) \(10+5-2\times1=13\) e) \(10\times(5-2)+1=31\)

- Identify the sequence of last digits that repeats. - Count how many terms are in one complete cycle. - Use the position of exponent \(50\) within the repeating cycle.

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 \(4\).

- How many small squares are in the new row and column? - Their total equals the difference of the two square areas. - The lengths of the two sides differ by one.

1. The new row and column contain \(n+(n+1)=2n+1\) squares. Hence \(2n+1=51\). 2. Subtract \(1\) and divide by \(2\): \(n=25\). The side lengths are \(25\) and \(26\). 3. The squares are \(25^2=625\) and \(26^2=676\). Their difference is \(676-625=51\).

a) Side lengths: \(25\) and \(26\) b) Perfect squares: \(625\) and \(676\); difference: \(51\)