Square and cube roots worksheet - Free Printable

Problem Analysis:

The image contains three sections (A, B, and C) with problems involving powers, square roots, and cube roots. Below is a detailed solution for each section.

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Section A: Powers

#### Instructions:
Find the value of each expression by calculating the power.

#### Solutions:
1. \( 2^4 \)
\( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)
Answer: \( 16 \)

2. \( -3^2 \)
Note: The negative sign is not part of the base. This means \( -3^2 = -(3^2) \).
\( 3^2 = 9 \), so \( -3^2 = -9 \).
Answer: \( -9 \)

3. \( 4^3 \)
\( 4^3 = 4 \times 4 \times 4 = 64 \)
Answer: \( 64 \)

4. \( (-5)^2 \)
Here, the negative sign is part of the base.
\( (-5)^2 = (-5) \times (-5) = 25 \)
Answer: \( 25 \)

5. \( 10^3 \)
\( 10^3 = 10 \times 10 \times 10 = 1000 \)
Answer: \( 1000 \)

6. \( 12^2 \)
\( 12^2 = 12 \times 12 = 144 \)
Answer: \( 144 \)

7. \( (-8)^3 \)
\( (-8)^3 = (-8) \times (-8) \times (-8) = -512 \)
Answer: \( -512 \)

8. \( 7^2 \)
\( 7^2 = 7 \times 7 = 49 \)
Answer: \( 49 \)

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Section B: Square Roots

#### Instructions:
Find the square root of each number. If the square root is not possible (e.g., the number is negative), write "IMPOSSIBLE."

#### Solutions:
11. \( \sqrt{16} \)
\( \sqrt{16} = 4 \) (since \( 4^2 = 16 \))
Answer: \( 4 \)

12. \( \sqrt{25} \)
\( \sqrt{25} = 5 \) (since \( 5^2 = 25 \))
Answer: \( 5 \)

13. \( \sqrt{-100} \)
The square root of a negative number is not defined in the real number system.
Answer: IMPOSSIBLE

14. \( \sqrt{81} \)
\( \sqrt{81} = 9 \) (since \( 9^2 = 81 \))
Answer: \( 9 \)

15. \( \sqrt{144} \)
\( \sqrt{144} = 12 \) (since \( 12^2 = 144 \))
Answer: \( 12 \)

16. \( \sqrt{1600} \)
\( \sqrt{1600} = 40 \) (since \( 40^2 = 1600 \))
Answer: \( 40 \)

17. \( \sqrt{8100} \)
\( \sqrt{8100} = 90 \) (since \( 90^2 = 8100 \))
Answer: \( 90 \)

18. \( \sqrt{49} \)
\( \sqrt{49} = 7 \) (since \( 7^2 = 49 \))
Answer: \( 7 \)

19. \( \sqrt{121} \)
\( \sqrt{121} = 11 \) (since \( 11^2 = 121 \))
Answer: \( 11 \)

20. \( \sqrt{-64} \)
The square root of a negative number is not defined in the real number system.
Answer: IMPOSSIBLE

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Section C: Cube Roots

#### Instructions:
Find the cube root of each number. If the cube root is not possible, write "IMPOSSIBLE."

#### Solutions:
21. \( \sqrt[3]{8} \)
\( \sqrt[3]{8} = 2 \) (since \( 2^3 = 8 \))
Answer: \( 2 \)

22. \( \sqrt[3]{27} \)
\( \sqrt[3]{27} = 3 \) (since \( 3^3 = 27 \))
Answer: \( 3 \)

23. \( \sqrt[3]{125} \)
\( \sqrt[3]{125} = 5 \) (since \( 5^3 = 125 \))
Answer: \( 5 \)

24. \( \sqrt[3]{1000} \)
\( \sqrt[3]{1000} = 10 \) (since \( 10^3 = 1000 \))
Answer: \( 10 \)

25. \( \sqrt[3]{-512} \)
\( \sqrt[3]{-512} = -8 \) (since \( (-8)^3 = -512 \))
Answer: \( -8 \)

26. \( \sqrt[3]{-125} \)
\( \sqrt[3]{-125} = -5 \) (since \( (-5)^3 = -125 \))
Answer: \( -5 \)

27. \( \sqrt[3]{64} \)
\( \sqrt[3]{64} = 4 \) (since \( 4^3 = 64 \))
Answer: \( 4 \)

28. \( \sqrt[3]{343} \)
\( \sqrt[3]{343} = 7 \) (since \( 7^3 = 343 \))
Answer: \( 7 \)

29. \( \sqrt[3]{729} \)
\( \sqrt[3]{729} = 9 \) (since \( 9^3 = 729 \))
Answer: \( 9 \)

30. \( \sqrt[3]{216} \)
\( \sqrt[3]{216} = 6 \) (since \( 6^3 = 216 \))
Answer: \( 6 \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
\text{A.} & 16, -9, 64, 25, 1000, 144, -512, 49 \\
\text{B.} & 4, 5, \text{IMPOSSIBLE}, 9, 12, 40, 90, 7, 11, \text{IMPOSSIBLE} \\
\text{C.} & 2, 3, 5, 10, -8, -5, 4, 7, 9, 6 \\
\end{array}
}
\]