Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem on the Exponents – Mixed Practice Worksheet 1 step by step, applying exponent rules. We'll simplify each expression and ensure no negative exponents appear in the final answer.

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Exponent Rules Used:

1. $ a^m \cdot a^n = a^{m+n} $
2. $ \frac{a^m}{a^n} = a^{m-n} $
3. $ (a^m)^n = a^{m \cdot n} $
4. $ (ab)^n = a^n b^n $
5. $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
6. $ a^{-n} = \frac{1}{a^n} $
7. $ a^0 = 1 $ (for $ a \neq 0 $)

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1. $ x^4 \cdot x^7 $

Add exponents:
$$
x^{4+7} = x^{11}
$$
✔ Answer: $ x^{11} $

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2. $ 2x^6 \cdot x^{11} $

Multiply coefficients and add exponents:
$$
2 \cdot x^{6+11} = 2x^{17}
$$
✔ Answer: $ 2x^{17} $

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3. $ \frac{g^{14}}{g^8} $

Subtract exponents:
$$
g^{14-8} = g^6
$$
✔ Answer: $ g^6 $

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4. $ (5p^4)^3 $

Apply power to both base and exponent:
$$
5^3 \cdot (p^4)^3 = 125 \cdot p^{12} = 125p^{12}
$$
✔ Answer: $ 125p^{12} $

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5. $ (3x^9y^3)^2 $

Square each part:
$$
3^2 \cdot (x^9)^2 \cdot (y^3)^2 = 9x^{18}y^6
$$
✔ Answer: $ 9x^{18}y^6 $

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6. $ (2m)^{-4} $

Negative exponent → move to denominator:
$$
\frac{1}{(2m)^4} = \frac{1}{2^4 m^4} = \frac{1}{16m^4}
$$
✔ Answer: $ \frac{1}{16m^4} $

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7. $ a^{-5} \cdot a^{-11} $

Add exponents:
$$
a^{-5 + (-11)} = a^{-16} = \frac{1}{a^{16}}
$$
✔ Answer: $ \frac{1}{a^{16}} $

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8. $ \frac{25x^9}{5x^{12}} $

Simplify coefficients and subtract exponents:
$$
\frac{25}{5} \cdot x^{9-12} = 5x^{-3} = \frac{5}{x^3}
$$
✔ Answer: $ \frac{5}{x^3} $

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9. $ (156u^{24}w^4)^0 $

Any nonzero expression raised to 0 is 1:
$$
= 1
$$
✔ Answer: $ 1 $

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10. $ (5x^5y^4)^2 $

Square each term:
$$
5^2 \cdot (x^5)^2 \cdot (y^4)^2 = 25x^{10}y^8
$$
✔ Answer: $ 25x^{10}y^8 $

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11. $ (-a^7b^5)^7 $

Apply exponent to each factor:
$$
(-1)^7 \cdot (a^7)^7 \cdot (b^5)^7 = -1 \cdot a^{49} \cdot b^{35} = -a^{49}b^{35}
$$
✔ Answer: $ -a^{49}b^{35} $

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12. $ (2x^5y)(3x^4y^7) $

Multiply coefficients and add exponents:
$$
(2 \cdot 3) \cdot x^{5+4} \cdot y^{1+7} = 6x^9y^8
$$
✔ Answer: $ 6x^9y^8 $

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13. $ (2y^4)^3(5y^6)^2 $

First expand each:
- $ (2y^4)^3 = 2^3 y^{12} = 8y^{12} $
- $ (5y^6)^2 = 25y^{12} $

Now multiply:
$$
8y^{12} \cdot 25y^{12} = 200y^{24}
$$
✔ Answer: $ 200y^{24} $

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14. $ \left(\frac{10x^9}{x^{18}}\right)^{-5} $

First simplify inside:
$$
\frac{10x^9}{x^{18}} = 10x^{9-18} = 10x^{-9}
$$

Now apply exponent $-5$:
$$
(10x^{-9})^{-5} = 10^{-5} \cdot x^{(-9)(-5)} = \frac{1}{10^5} \cdot x^{45} = \frac{x^{45}}{100000}
$$
✔ Answer: $ \frac{x^{45}}{100000} $

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15. $ \left(\frac{50x^8}{2x^3}\right)^2 $

Simplify inside first:
$$
\frac{50}{2} \cdot x^{8-3} = 25x^5
$$

Now square:
$$
(25x^5)^2 = 625x^{10}
$$
✔ Answer: $ 625x^{10} $

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16. $ (8k^7l^{12})^2(4k^{-4}j^3) $

First expand the squared term:
- $ (8k^7l^{12})^2 = 64k^{14}l^{24} $

Now multiply:
$$
64k^{14}l^{24} \cdot 4k^{-4}j^3 = (64 \cdot 4) \cdot k^{14 + (-4)} \cdot l^{24} \cdot j^3 = 256k^{10}l^{24}j^3
$$
✔ Answer: $ 256k^{10}l^{24}j^3 $

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17. $ \frac{a^8b^2}{4b} \cdot \frac{16a^9b^4}{a^3b^7} $

Break into parts:

First fraction: $ \frac{a^8b^2}{4b} = \frac{a^8b^{2-1}}{4} = \frac{a^8b}{4} $

Second fraction: $ \frac{16a^9b^4}{a^3b^7} = 16a^{9-3}b^{4-7} = 16a^6b^{-3} = \frac{16a^6}{b^3} $

Now multiply:
$$
\frac{a^8b}{4} \cdot \frac{16a^6}{b^3} = \frac{16}{4} \cdot a^{8+6} \cdot b^{1-3} = 4a^{14}b^{-2} = \frac{4a^{14}}{b^2}
$$
✔ Answer: $ \frac{4a^{14}}{b^2} $

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18. $ \frac{9x^7y^2}{xy} \cdot \frac{2x^6y^8}{2x^3y^4} $

Simplify each fraction:

First: $ \frac{9x^7y^2}{xy} = 9x^{7-1}y^{2-1} = 9x^6y $

Second: $ \frac{2x^6y^8}{2x^3y^4} = x^{6-3}y^{8-4} = x^3y^4 $

Now multiply:
$$
9x^6y \cdot x^3y^4 = 9x^{9}y^{5}
$$
✔ Answer: $ 9x^9y^5 $

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19. $ (3a^2b^6)^2(3a^{10}b^4)^3 $

Expand each:

- $ (3a^2b^6)^2 = 9a^4b^{12} $
- $ (3a^{10}b^4)^3 = 27a^{30}b^{12} $

Now multiply:
$$
9a^4b^{12} \cdot 27a^{30}b^{12} = (9 \cdot 27)a^{4+30}b^{12+12} = 243a^{34}b^{24}
$$
✔ Answer: $ 243a^{34}b^{24} $

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20. $ \left(\frac{5a^{10}b^5c^{22}}{15a^{-2}b^{14}c^{11}}\right)^3 $

First simplify inside:

Numerator: $ 5a^{10}b^5c^{22} $

Denominator: $ 15a^{-2}b^{14}c^{11} $

Divide:
$$
\frac{5}{15} \cdot a^{10 - (-2)} \cdot b^{5-14} \cdot c^{22-11} = \frac{1}{3} \cdot a^{12} \cdot b^{-9} \cdot c^{11}
= \frac{a^{12}c^{11}}{3b^9}
$$

Now cube the entire expression:
$$
\left(\frac{a^{12}c^{11}}{3b^9}\right)^3 = \frac{a^{36}c^{33}}{27b^{27}}
$$
✔ Answer: $ \frac{a^{36}c^{33}}{27b^{27}} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ x^{11} $ |
| 2 | $ 2x^{17} $ |
| 3 | $ g^6 $ |
| 4 | $ 125p^{12} $ |
| 5 | $ 9x^{18}y^6 $ |
| 6 | $ \frac{1}{16m^4} $ |
| 7 | $ \frac{1}{a^{16}} $ |
| 8 | $ \frac{5}{x^3} $ |
| 9 | $ 1 $ |
| 10 | $ 25x^{10}y^8 $ |
| 11 | $ -a^{49}b^{35} $ |
| 12 | $ 6x^9y^8 $ |
| 13 | $ 200y^{24} $ |
| 14 | $ \frac{x^{45}}{100000} $ |
| 15 | $ 625x^{10} $ |
| 16 | $ 256k^{10}l^{24}j^3 $ |
| 17 | $ \frac{4a^{14}}{b^2} $ |
| 18 | $ 9x^9y^5 $ |
| 19 | $ 243a^{34}b^{24} $ |
| 20 | $ \frac{a^{36}c^{33}}{27b^{27}} $ |

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