Multiplication Property of Exponents worksheets - Free Printable

Let's solve each problem step by step using the Multiplication Property of Exponents and other relevant exponent rules. Here are the key rules we'll use:

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🔷 Exponent Rules:

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

2. Power of a Product:
$ (ab)^n = a^n b^n $

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

4. Zero Exponent:
$ a^0 = 1 $ (for $ a \neq 0 $)

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

6. Coefficient and Variables:
Multiply coefficients separately and apply exponent rules to variables.

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Now, let's simplify each expression:

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1) $ 2m^2 \times 2m^3 $

- Coefficients: $ 2 \times 2 = 4 $
- Variables: $ m^2 \times m^3 = m^{2+3} = m^5 $
- ✔ Answer: $ 4m^5 $

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2) $ m^4 \times 2m^{-3} $

- Coefficient: $ 1 \times 2 = 2 $
- Variables: $ m^4 \times m^{-3} = m^{4 + (-3)} = m^1 = m $
- ✔ Answer: $ 2m $

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3) $ 5^2 \times 5^5 $

- Same base: $ 5^{2+5} = 5^7 $
- ✔ Answer: $ 5^7 $

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4) $ 4r^{-3} \times 2r^2 $

- Coefficients: $ 4 \times 2 = 8 $
- Variables: $ r^{-3} \times r^2 = r^{-3+2} = r^{-1} $
- So: $ 8r^{-1} $ or $ \frac{8}{r} $
- ✔ Answer: $ \frac{8}{r} $ or $ 8r^{-1} $

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5) $ 4n^4 \times 2n^{-3} $

- Coefficients: $ 4 \times 2 = 8 $
- Variables: $ n^4 \times n^{-3} = n^{4-3} = n^1 = n $
- ✔ Answer: $ 8n $

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6) $ x^2 \times x \times x^{-4} $

- All same base: $ x^{2+1+(-4)} = x^{-1} $
- ✔ Answer: $ x^{-1} $ or $ \frac{1}{x} $

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7) $ 2k^4 \times 4k $

- Coefficients: $ 2 \times 4 = 8 $
- Variables: $ k^4 \times k = k^{4+1} = k^5 $
- ✔ Answer: $ 8k^5 $

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8) $ 4v^3 \times vu^2 $

- Coefficients: $ 4 \times 1 = 4 $
- Variables:
- $ v^3 \times v = v^{3+1} = v^4 $
- $ u^2 $ remains
- So: $ 4v^4u^2 $
- ✔ Answer: $ 4v^4u^2 $

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9) $ (5^6)^0 $

- Any non-zero number to the power 0 is 1
- ✔ Answer: $ 1 $

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10) $ 2y^2 \times 3x $

- Coefficients: $ 2 \times 3 = 6 $
- Variables: $ y^2 \times x = xy^2 $
- ✔ Answer: $ 6xy^2 $

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11) $ 4(xy)^{-1} $

- $ (xy)^{-1} = \frac{1}{xy} $
- So: $ 4 \times \frac{1}{xy} = \frac{4}{xy} $
- ✔ Answer: $ \frac{4}{xy} $

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12) $ (-5x)^2 $

- Apply square to both: $ (-5)^2 \times x^2 = 25x^2 $
- ✔ Answer: $ 25x^2 $

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13) $ (3k^4)^4 $

- Apply power to both parts:
- $ 3^4 = 81 $
- $ (k^4)^4 = k^{4 \cdot 4} = k^{16} $
- ✔ Answer: $ 81k^{16} $

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14) $ (4a^3)^2 $

- $ 4^2 = 16 $
- $ (a^3)^2 = a^{6} $
- ✔ Answer: $ 16a^6 $

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15) $ (a^{-2} \times b^0)^3 $

- First: $ b^0 = 1 $
- So: $ a^{-2} \times 1 = a^{-2} $
- Now raise to 3: $ (a^{-2})^3 = a^{-6} $
- ✔ Answer: $ a^{-6} $ or $ \frac{1}{a^6} $

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16) $ (4r^0)^4 $

- $ r^0 = 1 $
- So: $ 4 \times 1 = 4 $
- $ 4^4 = 256 $
- ✔ Answer: $ 256 $

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17) $ (x^2)^0 $

- Any non-zero expression to the 0 power is 1
- ✔ Answer: $ 1 $

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18) $ (2^3)^2 $

- $ 2^3 = 8 $, then $ 8^2 = 64 $
- Or: $ (2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64 $
- ✔ Answer: $ 64 $

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19) $ (2x^2)^{-4} $

- Apply negative exponent: $ \frac{1}{(2x^2)^4} $
- $ (2x^2)^4 = 2^4 \cdot (x^2)^4 = 16x^8 $
- So: $ \frac{1}{16x^8} $
- ✔ Answer: $ \frac{1}{16x^8} $

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20) $ x^2 \times x^6 $

- Same base: $ x^{2+6} = x^8 $
- ✔ Answer: $ x^8 $

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21) $ -(2)^2 $

- This is negative of $ 2^2 $
- $ 2^2 = 4 $, so $ -(2)^2 = -4 $
- Note: Not $ (-2)^2 = 4 $ — here it's just $ -(2^2) $
- ✔ Answer: $ -4 $

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✔ Final Answers:

| # | Answer |
|---|--------|
| 1 | $ 4m^5 $ |
| 2 | $ 2m $ |
| 3 | $ 5^7 $ |
| 4 | $ \frac{8}{r} $ |
| 5 | $ 8n $ |
| 6 | $ \frac{1}{x} $ |
| 7 | $ 8k^5 $ |
| 8 | $ 4v^4u^2 $ |
| 9 | $ 1 $ |
|10 | $ 6xy^2 $ |
|11 | $ \frac{4}{xy} $ |
|12 | $ 25x^2 $ |
|13 | $ 81k^{16} $ |
|14 | $ 16a^6 $ |
|15 | $ \frac{1}{a^6} $ |
|16 | $ 256 $ |
|17 | $ 1 $ |
|18 | $ 64 $ |
|19 | $ \frac{1}{16x^8} $ |
|20 | $ x^8 $ |
|21 | $ -4 $ |

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Let me know if you'd like this as a printable PDF or want explanations for any specific ones!