Free Guide: Dividing Exponents Explained — Mashup Math - Free Printable

Let's solve each of these exponent problems step by step using the laws of exponents. The key rules we'll use are:

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. $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $
5. $ a^0 = 1 $ for $ a \neq 0 $

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1) $ \frac{3^5 \cdot 3^4}{3^8} $

- Numerator: $ 3^5 \cdot 3^4 = 3^{5+4} = 3^9 $
- So: $ \frac{3^9}{3^8} = 3^{9-8} = 3^1 = \boxed{3} $

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2) $ \frac{5^4 \cdot 5^3}{5^7} $

- Numerator: $ 5^{4+3} = 5^7 $
- $ \frac{5^7}{5^7} = 5^{7-7} = 5^0 = \boxed{1} $

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3) $ \frac{3^3 \cdot 3^4}{3^3} $

- Numerator: $ 3^{3+4} = 3^7 $
- $ \frac{3^7}{3^3} = 3^{7-3} = 3^4 = \boxed{81} $

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4) $ \frac{5}{5^3 \cdot 5^2} $

- Denominator: $ 5^3 \cdot 5^2 = 5^{5} $
- $ \frac{5}{5^5} = 5^{1-5} = 5^{-4} = \boxed{\frac{1}{5^4}} = \boxed{\frac{1}{625}} $

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5) $ \frac{6 \cdot 6^4}{6} $

- Numerator: $ 6^1 \cdot 6^4 = 6^5 $
- $ \frac{6^5}{6^1} = 6^{5-1} = 6^4 = \boxed{1296} $

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6) $ \frac{2^2 \cdot 2^4}{2} $

- Numerator: $ 2^{2+4} = 2^6 $
- $ \frac{2^6}{2^1} = 2^{6-1} = 2^5 = \boxed{32} $

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7) $ \frac{7^2 \cdot 7^4}{7} $

- Numerator: $ 7^{2+4} = 7^6 $
- $ \frac{7^6}{7^1} = 7^{5} = \boxed{16807} $

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8) $ \frac{64}{8^4} $

Note: $ 64 = 8^2 $, since $ 8^2 = 64 $

So: $ \frac{8^2}{8^4} = 8^{2-4} = 8^{-2} = \boxed{\frac{1}{8^2}} = \boxed{\frac{1}{64}} $

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9) $ \frac{8 \cdot 8^2}{8^3} $

- Numerator: $ 8^1 \cdot 8^2 = 8^3 $
- $ \frac{8^3}{8^3} = 8^{0} = \boxed{1} $

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10) $ \frac{4}{4^3 \cdot 4^2} $

- Denominator: $ 4^{3+2} = 4^5 $
- $ \frac{4}{4^5} = 4^{1-5} = 4^{-4} = \boxed{\frac{1}{4^4}} = \boxed{\frac{1}{256}} $

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11) $ \frac{7^2 \cdot 7^3}{7^2} $

- Numerator: $ 7^{2+3} = 7^5 $
- $ \frac{7^5}{7^2} = 7^{5-2} = 7^3 = \boxed{343} $

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12) $ \frac{8^3 \cdot 8^2}{8} $

- Numerator: $ 8^{3+2} = 8^5 $
- $ \frac{8^5}{8^1} = 8^{4} = \boxed{4096} $

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13) $ \frac{3^2}{3 \cdot 3^4} $

- Denominator: $ 3^1 \cdot 3^4 = 3^5 $
- $ \frac{3^2}{3^5} = 3^{2-5} = 3^{-3} = \boxed{\frac{1}{3^3}} = \boxed{\frac{1}{27}} $

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14) $ \frac{5^2 \cdot 5^4 \cdot 5^2}{5^3} $

- Numerator: $ 5^{2+4+2} = 5^8 $
- $ \frac{5^8}{5^3} = 5^{8-3} = 5^5 = \boxed{3125} $

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15) $ \frac{2 \cdot 2^2}{2^2} $

- Numerator: $ 2^1 \cdot 2^2 = 2^3 $
- $ \frac{2^3}{2^2} = 2^{3-2} = 2^1 = \boxed{2} $

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16) $ \frac{3^3}{3^2 \cdot 3^3} $

- Denominator: $ 3^{2+3} = 3^5 $
- $ \frac{3^3}{3^5} = 3^{3-5} = 3^{-2} = \boxed{\frac{1}{3^2}} = \boxed{\frac{1}{9}} $

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17) $ \frac{8^2 \cdot 8^2}{8^3} $

- Numerator: $ 8^{2+2} = 8^4 $
- $ \frac{8^4}{8^3} = 8^{4-3} = 8^1 = \boxed{8} $

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18) $ \frac{6^2 \cdot 6^4}{6^2} $

- Numerator: $ 6^{2+4} = 6^6 $
- $ \frac{6^6}{6^2} = 6^{6-2} = 6^4 = \boxed{1296} $

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19) $ \frac{8^2 \cdot 8^4}{8^4} $

- Numerator: $ 8^{2+4} = 8^6 $
- $ \frac{8^6}{8^4} = 8^{6-4} = 8^2 = \boxed{64} $

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20) $ \frac{2 \cdot 2^4}{2^4} $

- Numerator: $ 2^1 \cdot 2^4 = 2^5 $
- $ \frac{2^5}{2^4} = 2^{5-4} = 2^1 = \boxed{2} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ 3 $ |
| 2) | $ 1 $ |
| 3) | $ 81 $ |
| 4) | $ \frac{1}{625} $ |
| 5) | $ 1296 $ |
| 6) | $ 32 $ |
| 7) | $ 16807 $ |
| 8) | $ \frac{1}{64} $ |
| 9) | $ 1 $ |
| 10) | $ \frac{1}{256} $ |
| 11) | $ 343 $ |
| 12) | $ 4096 $ |
| 13) | $ \frac{1}{27} $ |
| 14) | $ 3125 $ |
| 15) | $ 2 $ |
| 16) | $ \frac{1}{9} $ |
| 17) | $ 8 $ |
| 18) | $ 1296 $ |
| 19) | $ 64 $ |
| 20) | $ 2 $ |

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