Multiplying Exponents (With Negatives) (A) - Free Printable

It looks like you've uploaded an image of a completed worksheet titled "Multiplying Exponents (A) Answers" from Math-Drills.com, which shows the simplified solutions to various exponent multiplication problems. Since the image contains already-solved answers, I’ll walk you through how each problem is solved, explaining the rules of exponents used.

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🔷 Key Rules Used:

1. Product of Powers Rule:
$ a^m \cdot a^n = a^{m+n} $

2. Negative Exponent Rule:
$ a^{-n} = \frac{1}{a^n} $

3. Power of a Power Rule:
$ (a^m)^n = a^{m \cdot n} $

4. Same Base, Combine Exponents:
When multiplying same bases, add exponents.

5. Even vs Odd Powers with Negatives:
$ (-a)^{\text{even}} = \text{positive} $,
$ (-a)^{\text{odd}} = \text{negative} $

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Now let's go through each problem step-by-step:

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1. $ 4^{-2} \cdot 4^{-1} $

- Same base: $ 4 $
- Add exponents: $ -2 + (-1) = -3 $
- So: $ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} $

✔ Answer: $ \frac{1}{4^3} $

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2. $ (-9)^5 \cdot (-9)^{-5} $

- Same base: $ (-9) $
- Add exponents: $ 5 + (-5) = 0 $
- Any nonzero number to the power of 0 is 1
- So: $ (-9)^0 = 1 $

✔ Answer: $ 1 $

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3. $ 2^{-1} \cdot 2^{-7} $

- Same base: $ 2 $
- Add exponents: $ -1 + (-7) = -8 $
- $ 2^{-8} = \frac{1}{2^8} = \frac{1}{256} $

✔ Answer: $ \frac{1}{2^8} $

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4. $ (-5)^6 \cdot (-5)^{-2} $

- Same base: $ (-5) $
- Add exponents: $ 6 + (-2) = 4 $
- So: $ (-5)^4 = 625 $ (since even power → positive)

✔ Answer: $ (-5)^4 $

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5. $ (-3)^5 \cdot (-3)^3 $

- Same base: $ (-3) $
- Add exponents: $ 5 + 3 = 8 $
- So: $ (-3)^8 $ (even power → positive)

✔ Answer: $ (-3)^8 $

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6. $ (-8)^{-1} \cdot (-8)^5 $

- Same base: $ (-8) $
- Add exponents: $ -1 + 5 = 4 $
- So: $ (-8)^4 $

✔ Answer: $ (-8)^4 $

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7. $ 7^1 \cdot 7^8 $

- Same base: $ 7 $
- Add exponents: $ 1 + 8 = 9 $
- So: $ 7^9 $

✔ Answer: $ 7^9 $

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8. $ 7^0 \cdot 7^{-9} $

- $ 7^0 = 1 $
- So: $ 1 \cdot 7^{-9} = 7^{-9} = \frac{1}{7^9} $

✔ Answer: $ \frac{1}{7^9} $

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9. $ 4^{-5} \cdot 4^2 $

- Same base: $ 4 $
- Add exponents: $ -5 + 2 = -3 $
- So: $ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} $

✔ Answer: $ \frac{1}{4^3} $

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10. $ (-4)^{-4} \cdot (-4)^{-1} $

- Same base: $ (-4) $
- Add exponents: $ -4 + (-1) = -5 $
- So: $ (-4)^{-5} = \frac{1}{(-4)^5} $

Note: $ (-4)^5 = -1024 $, so this is $ \frac{1}{-1024} $, but the answer is left as $ \frac{1}{(-4)^5} $

✔ Answer: $ \frac{1}{(-4)^5} $

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✔ Summary of Key Concepts:

- When multiplying powers with the same base, add exponents.
- Negative exponents mean reciprocals.
- Even powers of negative numbers are positive; odd powers are negative.
- Any nonzero number to the power of zero is 1.

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Let me know if you'd like these problems rewritten in a different format or want practice problems!