Multiplying Exponents With Different Bases and the Same Exponent ... - Free Printable

Let’s solve each problem one by one. We’ll use the rule:
When multiplying powers with the same exponent, you can multiply the bases and keep the exponent.
That is:
> \( a^n \cdot b^n = (a \cdot b)^n \)

But note: this only works if the exponents are the same. If they’re different, we can’t combine them this way — but in all these problems, the exponents *are* the same for each pair!

Also remember:
- A negative base raised to an even power becomes positive.
- A negative base raised to an odd power stays negative.
- Negative exponents mean “take the reciprocal”: \( x^{-n} = \frac{1}{x^n} \)

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Problem 1: \( 5^{-5} \cdot (-2)^{-5} \)

Same exponent: -5 → so combine bases:

\( [5 \cdot (-2)]^{-5} = (-10)^{-5} \)

Now, negative exponent:
\( (-10)^{-5} = \frac{1}{(-10)^5} \)

\( (-10)^5 = -100000 \) → because odd power keeps the sign.

So:
\( \frac{1}{-100000} = -\frac{1}{100000} \)

✔ Final Answer for #1: \( -\frac{1}{100000} \)

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Problem 2: \( (-3)^{-2} \cdot (-9)^{-2} \)

Same exponent: -2 → combine bases:

\( [(-3) \cdot (-9)]^{-2} = (27)^{-2} \)

Negative exponent:
\( 27^{-2} = \frac{1}{27^2} = \frac{1}{729} \)

✔ Final Answer for #2: \( \frac{1}{729} \)

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Problem 3: \( 4^{-8} \cdot 8^{-8} \)

Same exponent: -8 → combine bases:

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

Negative exponent:
\( 32^{-8} = \frac{1}{32^8} \)

We don’t need to compute 32^8 unless asked — it’s huge! But since the question says “simplify”, leaving as \( \frac{1}{32^8} \) is acceptable. However, sometimes they want it written as a single fraction. Let’s check if we can write 32 as a power of 2? Maybe not necessary here.

Actually, let’s see: 32 = 2^5, so 32^8 = (2^5)^8 = 2^{40}, so answer could be \( \frac{1}{2^{40}} \), but again — probably just leave as \( \frac{1}{32^8} \).

Wait — maybe the problem expects us to leave it as \( (32)^{-8} \)? But simplifying usually means no negative exponents.

So best simplified form: \( \frac{1}{32^8} \)

✔ Final Answer for #3: \( \frac{1}{32^8} \)

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Problem 4: \( 2^{-1} \cdot 8^{-1} \)

Same exponent: -1 → combine bases:

\( (2 \cdot 8)^{-1} = 16^{-1} = \frac{1}{16} \)

✔ Final Answer for #4: \( \frac{1}{16} \)

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Problem 5: \( 3^7 \cdot (-7)^7 \)

Same exponent: 7 → combine bases:

\( [3 \cdot (-7)]^7 = (-21)^7 \)

Odd power → stays negative.

We can leave it as \( (-21)^7 \), or compute? Probably leave as is unless specified.

But let’s think: is there a better way? No — this is simplified.

✔ Final Answer for #5: \( (-21)^7 \)

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Problem 6: \( (-7)^{-2} \cdot (-5)^{-2} \)

Same exponent: -2 → combine bases:

\( [(-7) \cdot (-5)]^{-2} = (35)^{-2} = \frac{1}{35^2} = \frac{1}{1225} \)

✔ Final Answer for #6: \( \frac{1}{1225} \)

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Problem 7: \( 7^3 \cdot (-6)^3 \)

Same exponent: 3 → combine bases:

\( [7 \cdot (-6)]^3 = (-42)^3 \)

Odd power → negative.

Leave as \( (-42)^3 \)

✔ Final Answer for #7: \( (-42)^3 \)

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Problem 8: \( 6^5 \cdot 9^5 \)

Same exponent: 5 → combine bases:

\( (6 \cdot 9)^5 = 54^5 \)

No negative exponents, so this is simplified.

✔ Final Answer for #8: \( 54^5 \)

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Problem 9: \( (-9)^{-9} \cdot 4^{-9} \)

Same exponent: -9 → combine bases:

\( [(-9) \cdot 4]^{-9} = (-36)^{-9} \)

Negative exponent:
\( (-36)^{-9} = \frac{1}{(-36)^9} \)

Odd power → denominator is negative → overall negative.

So: \( \frac{1}{-36^9} = -\frac{1}{36^9} \)

✔ Final Answer for #9: \( -\frac{1}{36^9} \)

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Problem 10: \( (-4)^{-9} \cdot 8^{-9} \)

Same exponent: -9 → combine bases:

\( [(-4) \cdot 8]^{-9} = (-32)^{-9} \)

Negative exponent:
\( (-32)^{-9} = \frac{1}{(-32)^9} \)

Odd power → denominator negative → overall negative.

So: \( \frac{1}{-32^9} = -\frac{1}{32^9} \)

✔ Final Answer for #10: \( -\frac{1}{32^9} \)

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Final Answers:

1. \( -\frac{1}{100000} \)
2. \( \frac{1}{729} \)
3. \( \frac{1}{32^8} \)
4. \( \frac{1}{16} \)
5. \( (-21)^7 \)
6. \( \frac{1}{1225} \)
7. \( (-42)^3 \)
8. \( 54^5 \)
9. \( -\frac{1}{36^9} \)
10. \( -\frac{1}{32^9} \)