Exponent product rule practice worksheet for middle school math.

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 $

---

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) $ 1^7 \cdot 1^{14} $

Same base: $ 1 $
$$
1^{7+14} = 1^{21}
$$
But $ 1^n = 1 $, so it simplifies to $ 1 $, but we are asked to write in exponential form.
✔ Answer: $ 1^{21} $

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

Any number to the power 0 is 1, so $ 8^0 = 1 $
$$
8^9 \cdot 1 = 8^9
$$
Or using 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 since we're writing in exponential form, $ x^0 $ is acceptable.
✔ Answer: $ x^0 $

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

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

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

Wait — this is #15, not #12. Let’s go in order.

Actually, looking at your list, you have numbers up to 25, but some are missing. Let me continue from where I left off.

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

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

---

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

Same base: $ 3 $
$$
3^{3+4} = 3^7
$$
✔ 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^5 \cdot x $

$ x = x^1 $
$$
x^{5+1} = x^6
$$
✔ Answer: $ x^6 $

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19) $ 8^3 \cdot 8^{11} $

$$
8^{3+11} = 8^{14}
$$
✔ Answer: $ 8^{14} $

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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^3 $

$$
y^{3+3} = y^6
$$
✔ Answer: $ y^6 $

---

22) $ m \cdot m^3 $

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

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23) $ 11^6 \cdot 11^5 $

$$
11^{6+5} = 11^{11}
$$
✔ Answer: $ 11^{11} $

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

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

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

$ 1 = 1^1 $, so:
$$
1^1 \cdot 1^{10} = 1^{11}
$$
But $ 1^{11} = 1 $, but again, we write in exponential form.
✔ Answer: $ 1^{11} $

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✔ Final Answers:

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

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🔍 Notes:

- Always check that bases are identical before applying the product rule.
- Negative exponents: add them normally.
- $ a^0 = 1 $ for any $ a \neq 0 $, so $ a^n \cdot a^{-n} = a^0 = 1 $.
- When multiplying variables, treat $ x $ as $ x^1 $.
- Be careful with negative bases: $ (-a)^m \cdot (-a)^n = (-a)^{m+n} $, but only if the base is exactly the same.

Let me know if you'd like this formatted as a printable worksheet!