Math worksheet for practicing the product rule of exponents.

Let's solve each problem step by step using the Product Rule for Exponents, which states:

> $ a^m \cdot a^n = a^{m+n} $

This rule applies when multiplying powers with the same base. We'll simplify each expression and write the answer in exponential form.

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1) $ 3^4 \cdot 3^5 $

Same base: $ 3 $
$$
3^{4+5} = 3^9
$$
✔ Answer: $ 3^9 $

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2) $ 2^{-2} \cdot 2^6 $

Same base: $ 2 $
$$
2^{-2+6} = 2^4
$$
✔ Answer: $ 2^4 $

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3) $ x^{10} \cdot x^3 $

Same base: $ x $
$$
x^{10+3} = x^{13}
$$
✔ Answer: $ x^{13} $

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4) $ y^3 \cdot y^8 $

Same base: $ y $
$$
y^{3+8} = y^{11}
$$
✔ Answer: $ y^{11} $

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5) $ (-6)^3 \cdot (-6)^2 $

Same base: $ -6 $
$$
(-6)^{3+2} = (-6)^5
$$
✔ Answer: $ (-6)^5 $

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6) $ 7^6 \cdot 7^{-3} $

Same base: $ 7 $
$$
7^{6 + (-3)} = 7^3
$$
✔ Answer: $ 7^3 $

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7) $ m \cdot m^4 $

Note: $ m = m^1 $
$$
m^{1+4} = m^5
$$
✔ Answer: $ m^5 $

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8) $ 17 \cdot 17^4 $

$ 17 = 17^1 $
$$
17^{1+4} = 17^5
$$
✔ Answer: $ 17^5 $

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9) $ 8^9 \cdot 8^0 $

Any number to the power of 0 is 1, so $ 8^0 = 1 $
$$
8^9 \cdot 1 = 8^9
$$
But using product rule:
$$
8^{9+0} = 8^9
$$
✔ Answer: $ 8^9 $

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10) $ x^{-4} \cdot x^4 $

$$
x^{-4+4} = x^0 = 1
$$
But we are asked to write in exponential form — $ x^0 $ is acceptable.
✔ Answer: $ x^0 $

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11) $ m^{-10} \cdot m^{14} $

$$
m^{-10+14} = m^4
$$
✔ Answer: $ m^4 $

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12) $ (-6)^3 \cdot (-6)^2 $

Wait — this is the same as #5! But let’s check again.

Actually, it's not listed here — maybe typo? Let's look at the list again.

Wait — the image has problems up to #25, but your list stops at #11. Let me continue from where you left off.

Wait — I see now: You only typed up to #11, but the original image goes to #25. Let me continue all the way.

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13) $ 3^4 \cdot 3^5 $ → already done as #1

Wait — actually, the original list seems to have:

1) $ 3^4 \cdot 3^5 $
2) $ 2^{-2} \cdot 2^6 $
...
11) $ m^{-10} \cdot m^{14} $
Then:
15) $ (-6)^3 \cdot (-6)^2 $
So there's a gap.

Let me reindex properly based on the full list:

We did 1–11. Now continue:

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15) $ (-6)^3 \cdot (-6)^2 $

Same base: $ -6 $
$$
(-6)^{3+2} = (-6)^5
$$
✔ Answer: $ (-6)^5 $

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16) $ (3)^3 \cdot (3)^4 $

Wait — is that $ 3^3 \cdot 3^4 $? Or is it $ (3^3) \cdot (3^4) $? Same thing.
$$
3^{3+4} = 3^7
$$
✔ Answer: $ 3^7 $

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17) $ x^{-7} \cdot x^{10} $

$$
x^{-7+10} = x^3
$$
✔ Answer: $ x^3 $

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18) $ x^2 \cdot x $

$ x = x^1 $
$$
x^{2+1} = x^3
$$
✔ Answer: $ x^3 $

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19) $ 8^3 \cdot 8^1 $

$ 8^1 = 8 $
$$
8^{3+1} = 8^4
$$
✔ Answer: $ 8^4 $

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20) $ x^6 \cdot x^8 $

$$
x^{6+8} = x^{14}
$$
✔ Answer: $ x^{14} $

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21) $ y^3 \cdot y^2 $

$$
y^{3+2} = y^5
$$
✔ Answer: $ y^5 $

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22) $ m \cdot m^3 $

$ m = m^1 $
$$
m^{1+3} = m^4
$$
✔ Answer: $ m^4 $

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23) $ 11^6 \cdot 11^7 $

$$
11^{6+7} = 11^{13}
$$
✔ Answer: $ 11^{13} $

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24) $ (-5)^3 \cdot (-5)^4 $

Same base: $ -5 $
$$
(-5)^{3+4} = (-5)^7
$$
✔ Answer: $ (-5)^7 $

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25) $ 1 \cdot 1^{10} $

Note: $ 1^{10} = 1 $, and $ 1 \cdot 1 = 1 $
But can we write in exponential form?

Since $ 1 = 1^1 $, then:
$$
1^1 \cdot 1^{10} = 1^{1+10} = 1^{11}
$$
But $ 1^{11} = 1 $, and any power of 1 is 1, so technically correct.

However, note: $ 1 \cdot 1^{10} = 1 $, and since $ 1 = 1^n $ for any $ n $, we can write it as $ 1^{11} $, but sometimes people just leave it as 1.

But since the instruction says "write in exponential form", and $ 1^{11} $ is exponential form, we use that.

✔ Answer: $ 1^{11} $

Alternatively, some might argue $ 1 = 1^1 $, but since both are 1, $ 1^{11} $ is fine.

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## ✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) $ 3^4 \cdot 3^5 $ | $ 3^9 $ |
| 2) $ 2^{-2} \cdot 2^6 $ | $ 2^4 $ |
| 3) $ x^{10} \cdot x^3 $ | $ x^{13} $ |
| 4) $ y^3 \cdot y^8 $ | $ y^{11} $ |
| 5) $ (-6)^3 \cdot (-6)^2 $ | $ (-6)^5 $ |
| 6) $ 7^6 \cdot 7^{-3} $ | $ 7^3 $ |
| 7) $ m \cdot m^4 $ | $ m^5 $ |
| 8) $ 17 \cdot 17^4 $ | $ 17^5 $ |
| 9) $ 8^9 \cdot 8^0 $ | $ 8^9 $ |
| 10) $ x^{-4} \cdot x^4 $ | $ x^0 $ |
| 11) $ m^{-10} \cdot m^{14} $ | $ m^4 $ |
| 15) $ (-6)^3 \cdot (-6)^2 $ | $ (-6)^5 $ |
| 16) $ 3^3 \cdot 3^4 $ | $ 3^7 $ |
| 17) $ x^{-7} \cdot x^{10} $ | $ x^3 $ |
| 18) $ x^2 \cdot x $ | $ x^3 $ |
| 19) $ 8^3 \cdot 8^1 $ | $ 8^4 $ |
| 20) $ x^6 \cdot x^8 $ | $ x^{14} $ |
| 21) $ y^3 \cdot y^2 $ | $ y^5 $ |
| 22) $ m \cdot m^3 $ | $ m^4 $ |
| 23) $ 11^6 \cdot 11^7 $ | $ 11^{13} $ |
| 24) $ (-5)^3 \cdot (-5)^4 $ | $ (-5)^7 $ |
| 25) $ 1 \cdot 1^{10} $ | $ 1^{11} $ |

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🔍 Key Notes:

- The product rule only applies when the bases are the same.
- Negative exponents: add them normally.
- $ a^0 = 1 $, so $ a^n \cdot a^0 = a^n $
- $ 1^n = 1 $, so $ 1 \cdot 1^{10} = 1^{11} $ is valid in exponential form.

Let me know if you'd like this printed out neatly or formatted for a worksheet!