Cube Root Calculations

• 1. What is the value of $\sqrt[3]{-8}$?

  • a) -4 b) -2 c) 2 d) 4

• 2. Find $\sqrt[3]{-27}$.

  • a) -9 b) -3 c) 3 d) 9

• 3. What is the cube root of 1728?

  • a) 10 b) 11 c) 12 d) 13

• 10. What is the cube root of 729?

  • a) 7 b) 8 c) 9 d) 10

Simplifying Expressions

• 4. Simplify $\sqrt[3]{8 \times 27}$.

  • a) 6 b) 12 c) 18 d) 24

• 5. Find the value of $\sqrt[3]{64 \times 125}$.

  • a) 10 b) 15 c) 20 d) 25

• 6. Simplify $\sqrt[3]{140 \times 2450}$.

  • a) 60 b) 70 c) 80 d) 90

• 7. Simplify $\sqrt[3]{108 \div 4}$.

  • a) 2 b) 3 c) 4 d) 5

• 8. Simplify $\sqrt[3]{512 \div 8}$.

  • a) 2 b) 3 c) 4 d) 6

Word Problems

• 9. The volume of a cube is $2197\text{ cm}^3$. What is the length of one side?

  • a) $11\text{ cm}$ b) $12\text{ cm}$ c) $13\text{ cm}$ d) $14\text{ cm}$

Step-by-Step Explanations

1. Value of $\sqrt[3]{-8}$

• Explanation: Since $(-2) \times (-2) \times (-2) = -8$, the cube root is $-2$.

• Correct Option: b

2. Value of $\sqrt[3]{-27}$

• Explanation: We look for a number that, when multiplied by itself three times, equals $-27$. $(-3)^3 = -27$.

• Correct Option: b

3. Cube root of 1728

• Explanation: You can use prime factorization or estimation. Since $10^3 = 1000$ and $20^3 = 8000$, the answer must be close to 10. $12 \times 12 \times 12 = 1728$.

• Correct Option: c

4. Simplify $\sqrt[3]{8 \times 27}$

• Explanation: Use the property $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$. Here, $\sqrt[3]{8} = 2$ and $\sqrt[3]{27} = 3$. Multiplying them gives $2 \times 3 = 6$.

• Correct Option: a

5. Value of $\sqrt[3]{64 \times 125}$

• Explanation: Similarly, $\sqrt[3]{64} = 4$ and $\sqrt[3]{125} = 5$. Multiplying these results gives $4 \times 5 = 20$.

• Correct Option: c

6. Simplify $\sqrt[3]{140 \times 2450}$

• Explanation: First, multiply the numbers: $140 \times 2450 = 343,000$. Since $\sqrt[3]{343} = 7$ and $\sqrt[3]{1000} = 10$, the answer is $7 \times 10 = 70$.

• Correct Option: b

7. Simplify $\sqrt[3]{108 \div 4}$

• Explanation: Perform the division inside the radical first: $108 \div 4 = 27$. The cube root of 27 is 3.

• Correct Option: b

8. Simplify $\sqrt[3]{512 \div 8}$

• Explanation: Divide 512 by 8 to get 64. The cube root of 64 is 4.

• Correct Option: c

9. Length of one side of a cube with volume $2197\text{ cm}^3$

• Explanation: The volume of a cube is $V = s^3$, so the side $s = \sqrt[3]{2197}$. Since $10^3 = 1000$ and the number ends in 7, we test 13. $13 \times 13 \times 13 = 2197$.

• Correct Option: c

10. Cube root of 729

• Explanation: By checking common cubes, $9 \times 9 \times 9 = 729$.

• Correct Option: c