Math worksheet for simplifying powers of exponents with negative bases and exponents.

Let's solve each expression step by step using the laws of exponents. We'll use the following key rules:

1. $(a^m)^n = a^{m \cdot n}$
2. $a^{-n} = \frac{1}{a^n}$
3. $(ab)^n = a^n b^n$ (not needed here, but good to know)
4. For negative bases: $(-a)^n$ depends on whether $n$ is even or odd.

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1. $((-5)^{-2})^5$

- First, simplify the inner exponent: $(-5)^{-2} = \frac{1}{(-5)^2} = \frac{1}{25}$
- Now raise that to the 5th power: $\left(\frac{1}{25}\right)^5 = \frac{1}{25^5}$
- But we can also apply the power rule directly:
$$
((-5)^{-2})^5 = (-5)^{-2 \cdot 5} = (-5)^{-10}
$$
- Since $(-5)^{-10} = \frac{1}{(-5)^{10}}$, and $(-5)^{10}$ is positive (even power):
$$
(-5)^{-10} = \frac{1}{5^{10}}
$$

✔ Answer: $\boxed{\frac{1}{5^{10}}}$

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2. $((-6)^{-1})^{-6}$

- Apply the power rule: $((-6)^{-1})^{-6} = (-6)^{(-1)(-6)} = (-6)^6$
- $(-6)^6 = 6^6$ since even power
- $6^6 = (2 \cdot 3)^6 = 2^6 \cdot 3^6 = 64 \cdot 729 = 46656$

✔ Answer: $\boxed{46656}$

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3. $(8^5)^1$

- Any number to the power of 1 is itself.
- So $(8^5)^1 = 8^5$

✔ Answer: $\boxed{8^5}$ (or $32768$ if evaluated)

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4. $((-6)^{-3})^4$

- Apply power rule: $(-6)^{-3 \cdot 4} = (-6)^{-12}$
- $(-6)^{-12} = \frac{1}{(-6)^{12}} = \frac{1}{6^{12}}$ (since even power)

✔ Answer: $\boxed{\frac{1}{6^{12}}}$

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5. $(4^7)^{-7}$

- Apply power rule: $4^{7 \cdot (-7)} = 4^{-49}$
- $4^{-49} = \frac{1}{4^{49}}$

✔ Answer: $\boxed{\frac{1}{4^{49}}}$

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6. $(8^{-6})^{-2}$

- Apply power rule: $8^{(-6)(-2)} = 8^{12}$

✔ Answer: $\boxed{8^{12}}$ (or $68719476736$ if evaluated)

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7. $((-9)^{-4})^{-6}$

- Apply power rule: $(-9)^{(-4)(-6)} = (-9)^{24}$
- Since 24 is even: $(-9)^{24} = 9^{24}$

✔ Answer: $\boxed{9^{24}}$

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8. $((-6)^9)^{-1}$

- Apply power rule: $(-6)^{9 \cdot (-1)} = (-6)^{-9}$
- $(-6)^{-9} = \frac{1}{(-6)^9}$
- Since $(-6)^9$ is negative (odd power): $= -\frac{1}{6^9}$

✔ Answer: $\boxed{-\frac{1}{6^9}}$

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9. $(2^{-5})^{-4}$

- Apply power rule: $2^{(-5)(-4)} = 2^{20}$

✔ Answer: $\boxed{2^{20}}$ (or $1,048,576$)

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10. $((-9)^3)^{-5}$

- Apply power rule: $(-9)^{3 \cdot (-5)} = (-9)^{-15}$
- $(-9)^{-15} = \frac{1}{(-9)^{15}}$
- Since $(-9)^{15}$ is negative (odd power): $= -\frac{1}{9^{15}}$

✔ Answer: $\boxed{-\frac{1}{9^{15}}}$

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Final Answers:

| Problem | Simplified Answer |
|--------|-------------------|
| 1 | $\frac{1}{5^{10}}$ |
| 2 | $46656$ |
| 3 | $8^5$ |
| 4 | $\frac{1}{6^{12}}$ |
| 5 | $\frac{1}{4^{49}}$ |
| 6 | $8^{12}$ |
| 7 | $9^{24}$ |
| 8 | $-\frac{1}{6^9}$ |
| 9 | $2^{20}$ |
| 10 | $-\frac{1}{9^{15}}$ |

Let me know if you'd like the numerical values for any of them!