Exponents Worksheets - Free Printable

Let's solve each of these exponent problems step by step using the laws of exponents. We'll simplify each expression and write the answers with positive exponents only.

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Exponent Rules Recap:

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

We’ll apply these rules one by one.

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1) $ \left( \frac{x^4 y^5}{x^2 y^3} \right)^{-2} $

Simplify inside first:

$$
\frac{x^4 y^5}{x^2 y^3} = x^{4-2} y^{5-3} = x^2 y^2
$$

Now raise to $-2$:

$$
(x^2 y^2)^{-2} = x^{-4} y^{-4}
$$

Convert to positive exponents:

$$
= \frac{1}{x^4 y^4}
$$

✔ Answer: $ \boxed{\frac{1}{x^4 y^4}} $

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2) $ (a^3 b^3)(ab^2)^2 $

First, expand $ (ab^2)^2 $:

$$
(ab^2)^2 = a^2 b^4
$$

Now multiply:

$$
a^3 b^3 \cdot a^2 b^4 = a^{3+2} b^{3+4} = a^5 b^7
$$

✔ Answer: $ \boxed{a^5 b^7} $

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3) $ \left( \frac{8m^3 n^{-4}}{2mn^{-3}} \right)^{-1} $

Simplify inside first:

$$
\frac{8m^3 n^{-4}}{2mn^{-3}} = \frac{8}{2} \cdot m^{3-1} \cdot n^{-4 - (-3)} = 4 m^2 n^{-1}
$$

Now raise to $-1$:

$$
(4 m^2 n^{-1})^{-1} = 4^{-1} m^{-2} n^{1} = \frac{n}{4 m^2}
$$

✔ Answer: $ \boxed{\frac{n}{4m^2}} $

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4) $ (5p^3 q^2)(2p^3 q)^2 $

First, expand $ (2p^3 q)^2 $:

$$
= 2^2 p^{6} q^2 = 4 p^6 q^2
$$

Now multiply:

$$
5p^3 q^2 \cdot 4p^6 q^2 = 20 p^{3+6} q^{2+2} = 20 p^9 q^4
$$

✔ Answer: $ \boxed{20p^9 q^4} $

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5) $ \frac{(4b^{-3})^2 (2b^4)}{4b^{-6}} $

First, expand numerator:

$$
(4b^{-3})^2 = 16 b^{-6}, \quad \text{so numerator: } 16 b^{-6} \cdot 2b^4 = 32 b^{-6+4} = 32 b^{-2}
$$

Denominator: $ 4b^{-6} $

Now divide:

$$
\frac{32 b^{-2}}{4 b^{-6}} = 8 b^{-2 - (-6)} = 8 b^{4}
$$

✔ Answer: $ \boxed{8b^4} $

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6) $ (a^{-2})^{-3}(a^4)^{-3} $

Apply power rule:

$$
(a^{-2})^{-3} = a^{(-2)(-3)} = a^6 \\
(a^4)^{-3} = a^{-12}
$$

Now multiply:

$$
a^6 \cdot a^{-12} = a^{6-12} = a^{-6} = \frac{1}{a^6}
$$

✔ Answer: $ \boxed{\frac{1}{a^6}} $

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7) $ \left( \frac{40x^2}{31y^{10}} \right)^{-3} $

Apply negative exponent:

$$
= \left( \frac{31y^{10}}{40x^2} \right)^3 = \frac{31^3 y^{30}}{40^3 x^6}
$$

Calculate powers:

- $ 31^3 = 29791 $
- $ 40^3 = 64000 $

So:

$$
= \frac{29791 y^{30}}{64000 x^6}
$$

✔ Answer: $ \boxed{\frac{29791 y^{30}}{64000 x^6}} $

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8) $ \left( \frac{2^{-3} a^{-4}}{x^5 a^{-5}} \right) (3x^{-4}) $

First, simplify the fraction:

$$
\frac{2^{-3} a^{-4}}{x^5 a^{-5}} = 2^{-3} x^{-5} a^{-4 - (-5)} = 2^{-3} x^{-5} a^{1}
$$

Now multiply by $ 3x^{-4} $:

$$
= 3 \cdot 2^{-3} x^{-5} x^{-4} a = \frac{3}{8} x^{-9} a
$$

Write with positive exponents:

$$
= \frac{3a}{8x^9}
$$

✔ Answer: $ \boxed{\frac{3a}{8x^9}} $

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9) $ \left( \frac{m^{-3} n^{-2}}{m^{-2} n^{-1}} \right)^{-4} $

Simplify inside:

$$
\frac{m^{-3} n^{-2}}{m^{-2} n^{-1}} = m^{-3 - (-2)} n^{-2 - (-1)} = m^{-1} n^{-1}
$$

Now raise to $-4$:

$$
(m^{-1} n^{-1})^{-4} = m^{4} n^{4}
$$

✔ Answer: $ \boxed{m^4 n^4} $

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10) $ \frac{5p^3 q^{-2} r^{-4}}{3p^{-1} q^{-3} r^{-6}} $

Divide coefficients and subtract exponents:

$$
= \frac{5}{3} p^{3 - (-1)} q^{-2 - (-3)} r^{-4 - (-6)} = \frac{5}{3} p^4 q^1 r^2
$$

✔ Answer: $ \boxed{\frac{5}{3} p^4 q r^2} $

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11) $ \left( \frac{3x^{-3} y^2}{4y^{-2}} \right)^{-2} $

Simplify inside:

$$
\frac{3x^{-3} y^2}{4y^{-2}} = \frac{3}{4} x^{-3} y^{2 - (-2)} = \frac{3}{4} x^{-3} y^4
$$

Now raise to $-2$:

$$
\left( \frac{3}{4} x^{-3} y^4 \right)^{-2} = \left( \frac{3}{4} \right)^{-2} x^{6} y^{-8} = \left( \frac{4}{3} \right)^2 x^6 y^{-8} = \frac{16}{9} x^6 y^{-8}
$$

Convert to positive exponents:

$$
= \frac{16 x^6}{9 y^8}
$$

✔ Answer: $ \boxed{\frac{16x^6}{9y^8}} $

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12) $ (3x^2 z^{-4})(2xz^{-3})(6z^{-3})^{-2} $

Break it down:

- First two terms: $ 3x^2 z^{-4} \cdot 2x z^{-3} = 6 x^{3} z^{-7} $
- Now compute $ (6z^{-3})^{-2} = 6^{-2} z^{6} = \frac{1}{36} z^6 $

Now multiply:

$$
6 x^3 z^{-7} \cdot \frac{1}{36} z^6 = \frac{6}{36} x^3 z^{-1} = \frac{1}{6} x^3 z^{-1} = \frac{x^3}{6z}
$$

✔ Answer: $ \boxed{\frac{x^3}{6z}} $

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13) $ (4a^2 z^{-3})^2 (z^{-3})^4 (a^{-2})^{-4} $

Compute each part:

- $ (4a^2 z^{-3})^2 = 16 a^4 z^{-6} $
- $ (z^{-3})^4 = z^{-12} $
- $ (a^{-2})^{-4} = a^{8} $

Now combine:

$$
16 a^4 z^{-6} \cdot z^{-12} \cdot a^8 = 16 a^{12} z^{-18}
$$

Convert to positive exponents:

$$
= \frac{16 a^{12}}{z^{18}}
$$

✔ Answer: $ \boxed{\frac{16a^{12}}{z^{18}}} $

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14) $ \left( \frac{6x^{-2} y^3}{2xy^{-4}} \right)^{-3} $

Simplify inside:

$$
\frac{6x^{-2} y^3}{2xy^{-4}} = 3 x^{-3} y^{7}
$$

Now raise to $-3$:

$$
(3 x^{-3} y^7)^{-3} = 3^{-3} x^{9} y^{-21} = \frac{1}{27} x^9 y^{-21} = \frac{x^9}{27 y^{21}}
$$

✔ Answer: $ \boxed{\frac{x^9}{27y^{21}}} $

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15) $ \frac{(2x^{-2})^3 (4x^3 z^{-3})}{4z^{-1}} $

Simplify numerator:

- $ (2x^{-2})^3 = 8 x^{-6} $
- So numerator: $ 8x^{-6} \cdot 4x^3 z^{-3} = 32 x^{-3} z^{-3} $

Denominator: $ 4z^{-1} $

Now divide:

$$
\frac{32 x^{-3} z^{-3}}{4 z^{-1}} = 8 x^{-3} z^{-3 - (-1)} = 8 x^{-3} z^{-2} = \frac{8}{x^3 z^2}
$$

✔ Answer: $ \boxed{\frac{8}{x^3 z^2}} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{1}{x^4 y^4} $ |
| 2 | $ a^5 b^7 $ |
| 3 | $ \frac{n}{4m^2} $ |
| 4 | $ 20p^9 q^4 $ |
| 5 | $ 8b^4 $ |
| 6 | $ \frac{1}{a^6} $ |
| 7 | $ \frac{29791 y^{30}}{64000 x^6} $ |
| 8 | $ \frac{3a}{8x^9} $ |
| 9 | $ m^4 n^4 $ |
| 10 | $ \frac{5}{3} p^4 q r^2 $ |
| 11 | $ \frac{16x^6}{9y^8} $ |
| 12 | $ \frac{x^3}{6z} $ |
| 13 | $ \frac{16a^{12}}{z^{18}} $ |
| 14 | $ \frac{x^9}{27y^{21}} $ |
| 15 | $ \frac{8}{x^3 z^2} $ |

Let me know if you'd like this in a printable format or need explanations for any specific problem!