- Reverse the cube-root operation. - Write the required radicand as a perfect cube. - Match its digits with the given pattern.

1. Cube the right side: \(9^3=729\). 2. Therefore, the radicand must be \(729\). 3. The missing tens digit is \(2\).

\(2\)

- The small raised number tells you how many equal factors there are, not what to multiply by. - To check a cube root, put your proposed answer back into a cubing calculation. - Compare the starting and ending numbers when you perform both operations in order.

1. Cubing gives \(3^3=3\cdot3\cdot3=27\). 2. Taking the cube root reverses cubing, so \(\sqrt[3]{27}=3\).

\(3^3=27\) and \(\sqrt[3]{27}=3\). Cube roots undo cubing.

- Which whole numbers have cubes close to \(27\) or \(-27\)? - Remember that you need a number whose cube equals the whole value under the root. - Does cubing your proposed answer return \(-27\)?

1. Since \((-3)(-3)(-3)=-27\), the cube of \(-3\) is \(-27\). 2. Therefore \(\sqrt[3]{-27}=-3\). An odd number of negative factors gives a negative product.

\(\sqrt[3]{-27}=-3\); checking gives \((-3)^3=-27\).

- What must be true about a cube's volume if its edge length is a whole number? - Compare \(1440\) with nearby perfect cubes instead of trying to evaluate the cube root exactly.

1. A cube with a whole-number edge length has a perfect-cube volume. 2. The consecutive perfect cubes around \(1440\) are \(11^3=1331\) and \(12^3=1728\). 3. Since \(1331<1440<1728\), \(1440\) is not a perfect cube, so the edge length is not a whole number of centimeters.

No. Since \(11^3<1440<12^3\), the volume is not a perfect cube, so the edge length cannot be a whole number of centimeters.

- Rewrite each radicand as a third power, keeping track of its sign. - For the fraction, examine numerator and denominator separately. - Check each result by cubing it.

1. Since \((-0.5)^3=-0.125\), \(\sqrt[3]{-0.125}=-0.5\). 2. Since \(15^3=3375\), \(\sqrt[3]{3375}=15\). 3. Since \(64=4^3\) and \(729=9^3\), \(\sqrt[3]{\frac{64}{729}}=\frac{4}{9}\).

a) \(-0.5\) b) \(15\) c) \(\frac{4}{9}\)

- Look for perfect cubes in the numerator and denominator. - Rewrite the fraction as one quantity raised to the third power. - Use the inverse relationship between cubing and cube roots.

1. \(343=7^3\) and \(1000=10^3\). 2. Therefore, \(\frac{343}{1000}=\left(\frac{7}{10}\right)^3\). 3. A cube equation with a positive right side has the real solution \(x=\frac{7}{10}\).

\(x=\frac{7}{10}\)

- Recall consecutive small cubes around each value. - Compare uncertain values with neighboring perfect cubes. - Write the base for every value that matches exactly.

1. \(7^3=343\), so \(343\) is a perfect cube. 2. Since \(7^3=343<500<512=8^3\), \(500\) is not a perfect cube. 3. \(9^3=729\), so \(729\) is a perfect cube. 4. \(11^3=1331\), so \(1331\) is a perfect cube. 5. Since \(11^3=1331<1500<1728=12^3\), \(1500\) is not a perfect cube.

\(343=7^3\), \(729=9^3\), and \(1331=11^3\)

- Check both proposed values by cubing them. - Pay attention to the sign of an odd power. - Compare the solution pattern with, rather than copy it from, square equations.

1. \(4^3=64\). 2. \((-4)^3=-64\), not \(64\). 3. Cubing preserves the sign of a real number, so a positive cube has one positive real cube root. 4. Therefore, the only real solution is \(x=4\).

Nora incorrectly applied the square-equation pattern. The only real solution is \(x=4\).

- Isolate the cubic term first. - Recognize the resulting perfect cube. - Check the solution in the original equation.

1. Divide by \(8\): \(x^3=125\). 2. Since \(5^3=125\), \(x=5\).

\(x=5\)

- Count the equal extent of the building in each direction. - Relate a solid cube’s total count to repeated multiplication. - Reverse that relationship with a cube root.

1. The building has \(5\) cubes in each of \(3\) dimensions. 2. The total number is \(5^3=125\). 3. The number along one edge is \(\sqrt[3]{125}=5\).

a) \(125\) unit cubes b) \(\sqrt[3]{125}=5\)

- Compare the radicand with the cube of the whole number. - Use the fact that larger positive numbers have larger cubes. - Avoid approximating the cube root directly.

1. Cube \(9\): \(9^3=729\). 2. Since \(700<729\), taking cube roots preserves the order. 3. Therefore, \(\sqrt[3]{700}<9\).

\(\sqrt[3]{700}<9\)

- Estimate the whole-number cube roots near the interval. - Check consecutive cubes until the bounds are crossed. - Confirm that only one cube lies inside.

1. \(12^3=1728\), which is below \(1800\). 2. \(13^3=2197\), which lies between \(1800\) and \(2200\). 3. \(14^3=2744\), which is above \(2200\). 4. Therefore, the unique perfect cube is \(2197\), with cube root \(13\).

\(2197=13^3\), so \(\sqrt[3]{2197}=13\).

- Identify which operation occurs first in each expression. - Use the inverse relationship between cubing and cube roots. - Check whether each result matches the original number inside the first operation.

1. \(A=18\) because taking the cube root reverses cubing. 2. \(\sqrt[3]{216}=6\), so \(B=6^3=216\). 3. Cubing and taking a real cube root are inverse operations. Applying one and then the other returns the original real number.

\(A=18\) and \(B=216\). Cubing and taking a real cube root undo each other for every real number.

- Compare the value of each side rather than assuming the root operation distributes. - The cube roots of the perfect cubes on the right can be evaluated exactly. - Check a proposed cube-root value by cubing it.

1. The left side is \(\sqrt[3]{35}\). 2. The right side is \(2+3=5\). 3. Since \(5^3=125\ne35\), \(\sqrt[3]{35}\ne5\). 4. Therefore, cube roots do not distribute over addition.

The claim is false. \(\sqrt[3]{35}\ne5\), so cube roots do not distribute over addition.

- Cube the fractional edge exactly. - Apply the inverse operation to the resulting volume. - Track the different units for length and volume.

1. The volume is \(\left(\frac{3}{2}\right)^3=\frac{27}{8}\,\text{ft}^3\). 2. Recovering the edge gives \(\sqrt[3]{\frac{27}{8}}=\frac{3}{2}\,\text{ft}\).

a) \(\frac{27}{8}\,\text{ft}^3\) b) \(\sqrt[3]{\frac{27}{8}}=\frac{3}{2}\,\text{ft}\)

- Recognize each factor as a perfect cube. - Combine the bases before evaluating the cube root. - Check the result by cubing it.

1. Rewrite \(8=2^3\) and \(343=7^3\). 2. Then \(8\cdot343=(2\cdot7)^3\). 3. Therefore, \(\sqrt[3]{8\cdot343}=2\cdot7=14\).

\(14\)

- Isolate the cubic term. - Recognize the resulting perfect cube. - Verify the solution in the original equation.

1. Subtract \(37\): \(x^3=27\). 2. Since \(3^3=27\), \(x=3\).

\(x=3\)

- Which operation undoes cubing a side length? - Every face of a cube is a square with the same side length. - Find a face's perimeter, not its area.

1. The edge length \(s\) satisfies \(s^3=512\). 2. Since \(8^3=512\), \(s=8\,\text{cm}\). 3. A square face has perimeter \(4\cdot8=32\,\text{cm}\).

a) \(8\,\text{cm}\) b) \(32\,\text{cm}\)

- List the perfect cubes that have exactly two digits. - Apply the digit condition to those few candidates. - State both the cube and its base.

1. The two-digit perfect cubes are \(27=3^3\) and \(64=4^3\). 2. The digit sum of \(27\) is \(9\). 3. The digit sum of \(64\) is \(10\). 4. Therefore, the number is \(64\), and its cube root is \(4\).

\(64\), with cube root \(4\)

- Check whether the numerator is a perfect cube. - Do the same for the denominator. - Verify the quotient by cubing it.

1. Both the numerator and denominator are perfect cubes: \(8=2^3\) and \(125=5^3\). 2. Therefore \(\sqrt[3]{\frac{8}{125}}=\frac{2}{5}\). 3. Check: \(\left(\frac{2}{5}\right)^3=\frac{8}{125}\).

\(\sqrt[3]{\frac{8}{125}}=\frac{2}{5}\), since \(\left(\frac{2}{5}\right)^3=\frac{8}{125}\).

- Read each number in the plan as a stack height. - Add the stack heights for each collection. - What total could be arranged into equally long rows, columns, and layers?

1. Plan a) has stack heights \(3,1,2,2\), giving \(3+1+2+2=8\) unit cubes. 2. Plan b) has stack heights \(1,3,2,3\), giving \(1+3+2+3=9\) unit cubes. 3. Since \(8=2^3\), collection a) can be rearranged into a solid cube with \(2\) units along each edge. The number \(9\) is not a perfect cube.

Collection a); \(2\) cubes along each edge.

- Which whole-number powers give \(64\)? - Work backward from each power to its base. - Why might two different operations, both starting with \(64\), give different answers?

1. Since \(8^2=64\), \(\sqrt{64}=8\). 2. Since \(4^3=64\), \(\sqrt[3]{64}=4\). 3. The square root reverses squaring, while the cube root reverses cubing, so the values need not match.

\(\sqrt{64}=8>4=\sqrt[3]{64}\). A square root is a number whose square gives the radicand; a cube root is a number whose cube gives it.

- Use the relationship between a cube's volume and its edge length. - Find two consecutive perfect cubes that enclose \(60\). - Test the midpoint of the two possible whole-number estimates.

1. The edge length is \(\sqrt[3]{60}\,\text{dm}\). 2. Since \(3^3=27\) and \(4^3=64\), \(27<60<64\). Therefore, the edge length is between \(3\,\text{dm}\) and \(4\,\text{dm}\). 3. The midpoint between \(3\) and \(4\) is \(3.5\). Since \(3.5^3=42.875<60\), \(\sqrt[3]{60}>3.5\). 4. Therefore, the edge length is closer to \(4\,\text{dm}\) than to \(3\,\text{dm}\).

a) The edge length is between \(3\,\text{dm}\) and \(4\,\text{dm}\). b) \(4\,\text{dm}\) is the better estimate because \(3.5^3=42.875<60\), so \(\sqrt[3]{60}>3.5\).

- Take the cube root of each volume. - Compare the two edge lengths. - You can also find the cube root of the volume ratio.

1. The smaller cube's edge length is \(\sqrt[3]{8}=2\,\text{dm}\). 2. The larger cube's edge length is \(\sqrt[3]{216}=6\,\text{dm}\). 3. The ratio is \(6\div2=3\), so the larger cube's edge is \(3\) times as long. 4. Equivalently, the volume ratio is \(216\div8=27=3^3\), so the edge-length ratio is \(\sqrt[3]{27}=3\).

The larger cube's edge length is \(3\) times the smaller cube's edge length, so the ratio is \(3{:}1\).

- Convert the volume to cubic centimeters. - Find the cube root of the volume. - A cube has six congruent square faces.

1. Convert the volume: \(0.008\,\text{m}^3=8000\,\text{cm}^3\). 2. The edge length is \(\sqrt[3]{8000}=20\,\text{cm}\). 3. The surface area is \(6s^2=6(20^2)=2400\,\text{cm}^2\).

a) The edge length is \(20\,\text{cm}\). b) The surface area is \(2400\,\text{cm}^2\).

- Use the volume of a cube to determine its edge. - Once the edge is known, find the area of one face. - Account for every congruent face.

1. The edge length \(s\) satisfies \(s^3=2744\). 2. Since \(14^3=2744\), \(s=14\,\text{ft}\). 3. One face has area \(14^2=196\,\text{ft}^2\). 4. The total surface area is \(6\cdot196=1176\,\text{ft}^2\).

a) \(14\,\text{ft}\) b) \(1176\,\text{ft}^2\)

- Add every visible stack height in the plan. - Compare the total with familiar perfect cubes. - Use a cube root to determine the edge of a rearranged solid cube.

1. Add the stack heights: \(1+2+3+4+5+4+3+2+3=27\). 2. Since \(27=3^3\), the total is a perfect cube. 3. The cubes can be rearranged into a solid cube with edge length \(\sqrt[3]{27}=3\) unit cubes.

a) \(27\) cubes b) Yes. The new cube has edge length \(3\) unit cubes.

- Relate a change in one dimension to the change in volume. - Write an equation for the unknown scale factor. - Reverse the third power.

1. If the edge-length scale factor is \(k\), the volume scale factor is \(k^3\). 2. Therefore, \(k^3=216\). 3. Since \(\sqrt[3]{216}=6\), \(k=6\).

The edge length is enlarged by a factor of \(6\).

- Find the perfect cubes immediately below and above the total. - Compute the change needed in each direction. - Choose the smaller change.

1. The neighboring perfect cubes are \(9^3=729\) and \(10^3=1000\). 2. Removing to reach \(729\) requires \(900-729=171\) cubes. 3. Adding to reach \(1000\) requires \(1000-900=100\) cubes. 4. Since \(100<171\), add \(100\) cubes.

Add \(100\) cubes to make \(1000=10^3\).

- Look for a decimal, whole number, and larger number that are all perfect cubes. - Compare corresponding edge lengths in order. - Relate the volume scale factor to a scale factor applied three times.

1. \(\sqrt[3]{0.008}=0.2\), \(\sqrt[3]{8}=2\), and \(\sqrt[3]{8000}=20\). 2. The edge lengths are \(0.2\,\text{cm}\), \(2\,\text{cm}\), and \(20\,\text{cm}\). 3. Each factor of \(1000=10^3\) in volume produces a factor of \(10\) in edge length.

a) \(0.2\,\text{cm}\), \(2\,\text{cm}\), and \(20\,\text{cm}\) b) The edge length is multiplied by \(10\).

- Find the number whose third power equals the volume. - Keep the original length unit while taking the cube root. - Convert the resulting linear measurement, not the volume.

1. The edge length \(s\) satisfies \(s^3=15{,}625\). 2. Since \(25^3=15{,}625\), \(s=25\,\text{mm}\). 3. Since \(10\,\text{mm}=1\,\text{cm}\), \(25\,\text{mm}=2.5\,\text{cm}\).

a) \(25\,\text{mm}\) b) \(2.5\,\text{cm}\)

- Rewrite the mixed number as an improper fraction. - Look for perfect cubes in the numerator and denominator. - Interpret the cube root as a length.

1. Convert the mixed number: \(1\frac{61}{64}=\frac{125}{64}\). 2. Since \(125=5^3\) and \(64=4^3\), \(\frac{125}{64}=\left(\frac{5}{4}\right)^3\). 3. The edge length is \(\frac{5}{4}\,\text{in}\).

\(\frac{5}{4}\,\text{in}\)

- Recover the original edge from the total number of unit cubes. - An outer layer affects both ends of every dimension. - Compare the original and remaining perfect cubes to find the removed amount.

1. a) \(\sqrt[3]{1000}=10\), so the original edge is \(10\) unit cubes long. 2. b) Removing one unit from both ends of each dimension leaves edge length \(10-2=8\). 3. The remaining cube contains \(8^3=512\) unit cubes. 4. c) The number removed is \(1000-512=488\).

a) \(10\) unit cubes b) \(512\) unit cubes remain. c) \(488\) unit cubes were removed.

- Work backward from a whole-number cube root. - Represent the fraction as a whole-number cube. - Use the upper bound to limit the possible cube roots.

1. Write \(\frac{n}{8}=m^3\), where \(m\) is a positive whole number. 2. Then \(n=8m^3\). 3. For \(m=1\), \(n=8\). For \(m=2\), \(n=64\). 4. For \(m=3\), \(n=216\), which exceeds \(100\). 5. Therefore, the valid values are \(8\) and \(64\).

\(n=8\) and \(n=64\)

- Examine the exponents rather than multiplying the factors. - Think about how cubing affects every prime exponent. - Divide valid exponents by \(3\) to obtain the cube root.

1. A perfect cube has exponents that are multiples of \(3\) in its prime factorization. 2. In \(A\), the exponents \(6\) and \(3\) are multiples of \(3\), so \(A\) is a perfect cube. 3. \(\sqrt[3]{A}=2^{6/3}\cdot3^{3/3}=2^2\cdot3=12\). 4. In \(B\), the exponent \(4\) is not a multiple of \(3\), so \(B\) is not a perfect cube.

a) \(A\) b) \(\sqrt[3]{A}=12\) c) \(B\) is not a perfect cube because the exponent of \(2\) is not divisible by \(3\).

- Focus on the exponents of the prime factors. - Determine what pattern cubing creates in those exponents. - Choose the least positive exponent that fits the pattern.

1. In a perfect cube, every prime exponent is a multiple of \(3\). 2. The exponent \(6\) is already a multiple of \(3\). 3. The smallest positive multiple of \(3\) for \(m\) is \(3\). 4. The cube root is \(2^{3/3}\cdot3^{6/3}=2\cdot3^2\).

\(m=3\), and the cube root is \(2\cdot3^2\).

- Test what remains after dividing by each possible coefficient. - The remaining value must be a perfect cube for the solution to be a positive integer. - Check the selected pair in the original equation.

1. Dividing by each candidate gives \(x^3=625\), \(125\), \(25\), or \(5\). 2. Only \(125\) is a perfect cube because \(125=5^3\). 3. This occurs when \(c=5\), and the positive integer solution is \(x=5\).

\(c=5\) and \(x=5\)

- Find consecutive whole-number cubes inside the volume interval. - Apply the parity condition to the possible edge lengths. - Match the selected edge to its cube.

1. \(16^3=4096\) and \(17^3=4913\), and both volumes lie in the interval. 2. The even edge length is \(16\,\text{cm}\). 3. Therefore, the volume is \(4096\,\text{cm}^3\).

Edge length: \(16\,\text{cm}\) Volume: \(4096\,\text{cm}^3\)

- Look for a decimal with one place whose cube could produce three decimal places. - Build the third power in two multiplication steps. - Verify the exact product before stating the cube root.

1. \(2.5^2=6.25\). 2. \(6.25\cdot2.5=15.625\), so \((2.5)^3=15.625\). 3. Therefore, \(\sqrt[3]{15.625}=2.5\).

\(2.5\)

- Identify the integer cube roots of both boundary values. - Think about how cubes change as integers increase through zero. - Include every integer between the boundary inputs.

1. Since \((-3)^3=-27\) and \(2^3=8\), the cube bounds correspond to \(-3\le x\le2\). 2. Cubing preserves the order of real numbers. 3. The integer solutions are \(-3,-2,-1,0,1,2\).

\(-3,-2,-1,0,1,2\)

- Use the stack heights to total the cubes. - Compare the total with nearby perfect cubes. - The edge of the completed solid is the cube root of its total.

1. There are \(15\) stacks of height \(4\), so the total is \(15\cdot4=60\) cubes. 2. The next perfect cube is \(64=4^3\). 3. Add \(64-60=4\) cubes. 4. The new solid cube has edge length \(\sqrt[3]{64}=4\) unit cubes.

a) \(60\) cubes b) Add \(4\) cubes. c) Edge length \(4\) unit cubes

- Recognize each radicand as a nearby whole-number cube. - Evaluate the separate cube roots before comparing the expressions. - Find the difference only after identifying the greater value.

1. \(512=8^3\), \(2197=13^3\), and \(3375=15^3\). 2. Therefore \(A=8+13=21\) and \(B=15\). 3. \(A-B=21-15=6\), so \(A\) is greater by \(6\).

\(A\) is greater than \(B\) by \(6\).

- Consider a decimal less than \(1\) with two places. - Build its cube as a square times the original number. - Check that every decimal place matches the radicand.

1. \(0.75^2=0.5625\). 2. \(0.5625\cdot0.75=0.421875\). 3. Therefore, \((0.75)^3=0.421875\), so \(\sqrt[3]{0.421875}=0.75\).

\(0.75\)

- Isolate the cubic expression. - Recognize both factors as perfect cubes. - Combine their cube roots before multiplying large numbers.

1. Multiply by \(64\): \(x^3=27\cdot64\). 2. Since \(27=3^3\) and \(64=4^3\), \(x^3=(3\cdot4)^3=12^3\). 3. Therefore, \(x=12\).

\(x=12\)

- Continue the sequence by cubing the next whole number. - Subtract adjacent terms in order. - Compare each difference with the small perfect cubes.

1. The next perfect cube is \(5^3=125\). 2. The consecutive differences are \(8-1=7\), \(27-8=19\), \(64-27=37\), and \(125-64=61\). 3. None of \(7,19,37,61\) is a perfect cube.

a) \(125\) b) \(7,19,37,61\) c) No; none of the differences is a perfect cube.

- Convert both measurements to centimeter-based units. - Use a cube root to find Cube A's edge. - Use \(SA=6s^2\) to find Cube B's edge.

1. Convert Cube A's volume: \(0.008\,\text{dm}^3=8\,\text{cm}^3\). 2. Its edge length is \(\sqrt[3]{8}=2\,\text{cm}\). 3. For Cube B, \(6s^2=54\), so \(s^2=9\) and \(s=3\,\text{cm}\). 4. Cube B has the longer edge. Its volume is \(3^3=27\,\text{cm}^3\). 5. The volume difference is \(27-8=19\,\text{cm}^3\).

Cube B has the longer edge: \(3\,\text{cm}\), compared with \(2\,\text{cm}\) for Cube A. The volume difference is \(19\,\text{cm}^3\).

- Factor the number into primes. - Determine which exponents need to reach the next multiple of \(3\). - Use only the missing factors so the multiplier is least.

1. Factor \(54=2\cdot3^3\). 2. The exponent of \(3\) already fits a perfect cube. 3. The factor \(2\) needs two more factors of \(2\), so multiply by \(2^2=4\). 4. The product is \(54\cdot4=216=6^3\).

Multiply by \(4\). The product is \(216=6^3\).