Square Roots and Cube Roots Worksheet

1. What is √196?

2. What is ∛216?

3. Between which integers is √50?

4. Which statement verifies ∛343 = 7?

5. Which calculation verifies √(9 × 16) = √9 × √16?

6. Evaluate √324 and verify.

7. Evaluate ∛512 and verify.

8. Locate √115 between consecutive integers and state which it is closer to.

9. A square has area 288 cm². Between which whole numbers is its side length?

10. Estimate ∛90 between consecutive whole numbers, state which integer it is closer to, then check with a calculator.

11. Classify each answer as exact or approximate: √196 = 14 and √70 ≈ 8.37. Explain.

12. Verify √(4 × 49) = √4 × √49, then evaluate √64 + ∛125 × 3².

Answers and reasoning

1. Answer

14. 14² = 196.

2. Answer

6. 6³ = 216.

3. Answer

7 and 8. 7² = 49 and 8² = 64.

4. Answer

7³ = 343. A cube root is verified by cubing the proposed value.

5. Answer

√144 = 12 and 3 × 4 = 12. Both sides evaluate to 12, verifying the relationship for this numerical example.

6. Model response

√324 = 18 because 18² = 324.

7. Model response

∛512 = 8 because 8³ = 512.

8. Model response

10² = 100 and 11² = 121, so 10 < √115 < 11. It is closer to 11 because 115 is 6 below 121 but 15 above 100.

9. Model response

16² = 256 and 17² = 289, so the side length √288 is between 16 cm and 17 cm, very close to 17 cm.

10. Model response

4³ = 64 and 5³ = 125, so 4 < ∛90 < 5. It is closer to 4 because 90 is 26 above 64 and 35 below 125. A calculator gives ∛90 ≈ 4.48.

11. Model response

√196 = 14 is exact because 14² = 196. √70 ≈ 8.37 is approximate because 70 is not a perfect square and the decimal has been rounded.

12. Model response

Both sides of the root product equal 14: √196 = 14 and 2 × 7 = 14. For the expression, roots and powers first: 8 + 5 × 9 = 8 + 45 = 53.