Power Of A Power Worksheet, Rules, And Definition - Free Printable

Let's solve each problem using the Power of a Power Rule, which states:

> $(a^m)^n = a^{m \cdot n}$

This means when you have a power raised to another power, you multiply the exponents.

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Problem 1: $(2^2)^2$

$$
(2^2)^2 = 2^{2 \cdot 2} = 2^4 = 16
$$

✔ Answer: $16$

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Problem 2: $(8^8)^4$

$$
(8^8)^4 = 8^{8 \cdot 4} = 8^{32}
$$

✔ Answer: $8^{32}$

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Problem 3: $(4^{-2})^{-2}$

$$
(4^{-2})^{-2} = 4^{(-2) \cdot (-2)} = 4^4 = 256
$$

✔ Answer: $256$

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Problem 4: $(11^{-10})^{-4}$

$$
(11^{-10})^{-4} = 11^{(-10) \cdot (-4)} = 11^{40}
$$

✔ Answer: $11^{40}$

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Problem 5: $(18^{-3})^{-5}$

$$
(18^{-3})^{-5} = 18^{(-3) \cdot (-5)} = 18^{15}
$$

✔ Answer: $18^{15}$

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Problem 6: $(x^{-2})^{-4}$

$$
(x^{-2})^{-4} = x^{(-2) \cdot (-4)} = x^8
$$

✔ Answer: $x^8$

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Problem 7: $(x^6)^3$

$$
(x^6)^3 = x^{6 \cdot 3} = x^{18}
$$

✔ Answer: $x^{18}$

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Problem 8: $(x^{-5})^{-20}$

$$
(x^{-5})^{-20} = x^{(-5) \cdot (-20)} = x^{100}
$$

✔ Answer: $x^{100}$

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Problem 9: $(x^{-15})^{-3}$

$$
(x^{-15})^{-3} = x^{(-15) \cdot (-3)} = x^{45}
$$

✔ Answer: $x^{45}$

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Problem 10: $(x^{-50})^{-4}$

$$
(x^{-50})^{-4} = x^{(-50) \cdot (-4)} = x^{200}
$$

✔ Answer: $x^{200}$

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✔ Final Answers:

1. $16$
2. $8^{32}$
3. $256$
4. $11^{40}$
5. $18^{15}$
6. $x^8$
7. $x^{18}$
8. $x^{100}$
9. $x^{45}$
10. $x^{200}$

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🔍 Explanation Summary:

- The Power of a Power Rule: Multiply the exponents.
- When multiplying negative exponents: negative × negative = positive.
- Always simplify the exponent first, then write the final expression.

Let me know if you'd like these written in a formatted worksheet style!