Square Roots and Cube Roots | Interactive Worksheet - Free Printable

Problem Overview:

The task involves solving for square roots and cube roots of given numbers. The worksheet is divided into two sections:
1. Finding Square Roots: For example, the square root of 16 is 4 because \(4^2 = 16\).
2. Finding Cube Roots: For example, the cube root of 125 is 5 because \(5^3 = 125\).

Additionally, there are challenge questions at the end.

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Step-by-Step Solution:

#### Section 1: Finding Square Roots
We need to find the square root of each given number. The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\).

1. \(\sqrt{4} = 2\) because \(2^2 = 4\).
2. \(\sqrt{25} = 5\) because \(5^2 = 25\).
3. \(\sqrt{49} = 7\) because \(7^2 = 49\).
4. \(\sqrt{81} = 9\) because \(9^2 = 81\).
5. \(\sqrt{9} = 3\) because \(3^2 = 9\).
6. \(\sqrt{64} = 8\) because \(8^2 = 64\).
7. \(\sqrt{36} = 6\) because \(6^2 = 36\).
8. \(\sqrt{144} = 12\) because \(12^2 = 144\).
9. \(\sqrt{169} = 13\) because \(13^2 = 169\).
10. \(\sqrt{121} = 11\) because \(11^2 = 121\).
11. \(\sqrt{400} = 20\) because \(20^2 = 400\).
12. \(\sqrt{225} = 15\) because \(15^2 = 225\).

#### Section 2: Finding Cube Roots
We need to find the cube root of each given number. The cube root of a number \(x\) is a value \(y\) such that \(y^3 = x\).

13. \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
14. \(\sqrt[3]{512} = 8\) because \(8^3 = 512\).
15. \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
16. \(\sqrt[3]{1331} = 11\) because \(11^3 = 1331\).
17. \(\sqrt[3]{1331} = 11\) (repeated from above).
18. \(\sqrt[3]{1728} = 12\) because \(12^3 = 1728\).
19. \(\sqrt[3]{1331} = 11\) (repeated from above).
20. \(\sqrt[3]{729} = 9\) because \(9^3 = 729\).
21. \(\sqrt[3]{1000} = 10\) because \(10^3 = 1000\).

#### Challenge Questions
22. What number has a square root of 13?
- If the square root of a number is 13, then the number is \(13^2 = 169\).
- Answer: \(169\)

23. What number has a cube root of 6?
- If the cube root of a number is 6, then the number is \(6^3 = 216\).
- Answer: \(216\)

24. What is the square root of the square root of 81?
- First, find the square root of 81: \(\sqrt{81} = 9\).
- Next, find the square root of 9: \(\sqrt{9} = 3\).
- Answer: \(3\)

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Final Answers:

1. \(\sqrt{4} = 2\)
2. \(\sqrt{25} = 5\)
3. \(\sqrt{49} = 7\)
4. \(\sqrt{81} = 9\)
5. \(\sqrt{9} = 3\)
6. \(\sqrt{64} = 8\)
7. \(\sqrt{36} = 6\)
8. \(\sqrt{144} = 12\)
9. \(\sqrt{169} = 13\)
10. \(\sqrt{121} = 11\)
11. \(\sqrt{400} = 20\)
12. \(\sqrt{225} = 15\)
13. \(\sqrt[3]{8} = 2\)
14. \(\sqrt[3]{512} = 8\)
15. \(\sqrt[3]{27} = 3\)
16. \(\sqrt[3]{1331} = 11\)
17. \(\sqrt[3]{1331} = 11\)
18. \(\sqrt[3]{1728} = 12\)
19. \(\sqrt[3]{1331} = 11\)
20. \(\sqrt[3]{729} = 9\)
21. \(\sqrt[3]{1000} = 10\)
22. Challenge Question 1: \(169\)
23. Challenge Question 2: \(216\)
24. Challenge Question 3: \(3\)

Boxed Final Answer:
\[
\boxed{169, 216, 3}
\]