Division Property of Exponents worksheet with 21 simplification problems.

Let's solve each problem step by step using the Division Property of Exponents.

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🔷 Division Property of Exponents:

For any nonzero base $ a $ and integers $ m $ and $ n $:

$$
\frac{a^m}{a^n} = a^{m-n}
$$

Also, when you have a power of a quotient:

$$
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
$$

And for negative exponents:

$$
a^{-n} = \frac{1}{a^n}
$$

Now let’s simplify each expression:

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1) $ \frac{b^{-4}}{b^4} $

Use $ \frac{b^m}{b^n} = b^{m-n} $:

$$
b^{-4 - 4} = b^{-8} = \boxed{\frac{1}{b^8}}
$$

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2) $ \frac{5^2}{5^9} $

$$
5^{2-9} = 5^{-7} = \boxed{\frac{1}{5^7}}
$$

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3) $ \frac{5^6}{5^9} $

$$
5^{6-9} = 5^{-3} = \boxed{\frac{1}{5^3}} = \boxed{\frac{1}{125}}
$$

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4) $ \left(\frac{5}{3}\right)^3 $

Apply power to numerator and denominator:

$$
\frac{5^3}{3^3} = \frac{125}{27} = \boxed{\frac{125}{27}}
$$

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5) $ \frac{n^2}{n^3} $

$$
n^{2-3} = n^{-1} = \boxed{\frac{1}{n}}
$$

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6) $ \frac{n^2}{n} $

$$
n^{2-1} = n^1 = \boxed{n}
$$

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7) $ \frac{x}{x^5} $

$$
x^{1-5} = x^{-4} = \boxed{\frac{1}{x^4}}
$$

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8) $ \frac{2^{12}}{2^{11}} $

$$
2^{12-11} = 2^1 = \boxed{2}
$$

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9) $ \frac{2^{10}}{2^{11}} $

$$
2^{10-11} = 2^{-1} = \boxed{\frac{1}{2}}
$$

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10) $ \frac{5^8}{5^4} $

$$
5^{8-4} = 5^4 = \boxed{625}
$$

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11) $ \left(\frac{1}{5}\right)^3 $

$$
\frac{1^3}{5^3} = \frac{1}{125} = \boxed{\frac{1}{125}}
$$

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12) $ \left(\frac{1}{5}\right)^4 $

$$
\frac{1^4}{5^4} = \frac{1}{625} = \boxed{\frac{1}{625}}
$$

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13) $ \frac{3^4 \cdot x^9}{3^1 \cdot x^8} $

Break into parts:

- $ \frac{3^4}{3^1} = 3^{4-1} = 3^3 = 27 $
- $ \frac{x^9}{x^8} = x^{9-8} = x^1 = x $

So total: $ \boxed{27x} $

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14) $ \frac{a^{17}}{a^{14}} $

$$
a^{17-14} = a^3 = \boxed{a^3}
$$

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15) $ \frac{a^6}{a^{14}} $

$$
a^{6-14} = a^{-8} = \boxed{\frac{1}{a^8}}
$$

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16) $ \frac{x^3}{x^5} $

$$
x^{3-5} = x^{-2} = \boxed{\frac{1}{x^2}}
$$

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17) $ \frac{x^{12}}{x^6} $

$$
x^{12-6} = x^6 = \boxed{x^6}
$$

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18) $ \frac{x^{12}}{x^9} $

$$
x^{12-9} = x^3 = \boxed{x^3}
$$

*(Note: Problem #18 is labeled as "12)" in the image — likely a typo. We’ll assume it's #18.)*

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19) $ \frac{3}{3^6} $

$$
3^{1-6} = 3^{-5} = \boxed{\frac{1}{3^5}} = \boxed{\frac{1}{243}}
$$

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20) $ \frac{c^{12}}{c^5} $

$$
c^{12-5} = c^7 = \boxed{c^7}
$$

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21) $ \frac{3}{3^5} $

Wait — this is same as #19, but labeled as #21. Let's check:

$$
\frac{3}{3^5} = 3^{1-5} = 3^{-4} = \boxed{\frac{1}{3^4}} = \boxed{\frac{1}{81}}
$$

But wait! Earlier (#19) was $ \frac{3}{3^6} = 3^{-5} = \frac{1}{243} $. So this one is different.

So #21 is $ \frac{3}{3^5} = 3^{1-5} = 3^{-4} = \boxed{\frac{1}{81}} $

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✔ Final Answers (Corrected):

| Problem | Answer |
|--------|--------|
| 1) | $ \frac{1}{b^8} $ |
| 2) | $ \frac{1}{5^7} $ |
| 3) | $ \frac{1}{125} $ |
| 4) | $ \frac{125}{27} $ |
| 5) | $ \frac{1}{n} $ |
| 6) | $ n $ |
| 7) | $ \frac{1}{x^4} $ |
| 8) | $ 2 $ |
| 9) | $ \frac{1}{2} $ |
| 10) | $ 625 $ |
| 11) | $ \frac{1}{125} $ |
| 12) | $ \frac{1}{625} $ |
| 13) | $ 27x $ |
| 14) | $ a^3 $ |
| 15) | $ \frac{1}{a^8} $ |
| 16) | $ \frac{1}{x^2} $ |
| 17) | $ x^6 $ |
| 18) | $ x^3 $ |
| 19) | $ \frac{1}{243} $ |
| 20) | $ c^7 $ |
| 21) | $ \frac{1}{81} $ |

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📌 Notes:

- Problems #19 and #21 both involve $ \frac{3}{3^n} $, but with different exponents.
- The original worksheet has a typo: Problem #18 is labeled as "12)", and #19 is labeled as "13)" — but based on numbering, we assumed correct sequence.

Let me know if you'd like this formatted as a printable answer key or need explanations for specific ones!