Worksheet on simplifying expressions using exponent rules.

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}}{2m^{-1} n^2} \right)^{-1} $

Simplify inside:

Numerator: $ 8m^3 n^{-4} $
Denominator: $ 2m^{-1} n^2 $

Divide:

$$
\frac{8}{2} \cdot m^{3 - (-1)} \cdot n^{-4 - 2} = 4 \cdot m^{4} \cdot n^{-6}
$$

So:

$$
\left(4 m^4 n^{-6}\right)^{-1} = 4^{-1} m^{-4} n^{6} = \frac{1}{4} m^{-4} n^6
$$

Convert to positive exponents:

$$
= \frac{n^6}{4 m^4}
$$

✔ Answer: $ \boxed{\frac{n^6}{4 m^4}} $

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4) $ (5p^4 q^{-1})(2p^3 q)^2 $

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

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

Now multiply:

$$
5p^4 q^{-1} \cdot 4 p^6 q^2 = 20 p^{4+6} q^{-1+2} = 20 p^{10} q^1
$$

✔ Answer: $ \boxed{20 p^{10} q} $

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5) $ \frac{(2k^{-1})^2 (2k)^2}{4k^{-2}} $

First, compute numerator:

- $ (2k^{-1})^2 = 4 k^{-2} $
- $ (2k)^2 = 4 k^2 $

Multiply:

$$
4k^{-2} \cdot 4k^2 = 16 k^{-2 + 2} = 16 k^0 = 16
$$

Denominator: $ 4k^{-2} $

So:

$$
\frac{16}{4k^{-2}} = 4 \cdot k^2 = 4k^2
$$

✔ Answer: $ \boxed{4k^2} $

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6) $ (a^{-2})^{-1}(a^3)^{-2} a^{-3} $

Simplify each part:

- $ (a^{-2})^{-1} = a^{(-2)(-1)} = a^2 $
- $ (a^3)^{-2} = a^{-6} $
- Then $ a^{-3} $

Now combine:

$$
a^2 \cdot a^{-6} \cdot a^{-3} = a^{2 - 6 - 3} = a^{-7}
$$

Convert to positive exponent:

$$
= \frac{1}{a^7}
$$

✔ Answer: $ \boxed{\frac{1}{a^7}} $

---

7) $ \left( \frac{24 x^2}{31 y^{10}} \right)^{-3} $

Apply negative exponent:

$$
= \left( \frac{31 y^{10}}{24 x^2} \right)^3
$$

Now distribute exponent:

$$
= \frac{31^3 y^{30}}{24^3 x^6}
$$

Calculate powers:

- $ 31^3 = 29791 $
- $ 24^3 = 13824 $

So:

$$
= \frac{29791 y^{30}}{13824 x^6}
$$

✔ Answer: $ \boxed{\frac{29791 y^{30}}{13824 x^6}} $

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8) $ \left( \frac{2^{-4} a^{-4}}{a^3 b^{-1}} \right) (3a^{-3}) $

First, simplify the fraction:

$$
\frac{2^{-4} a^{-4}}{a^3 b^{-1}} = 2^{-4} a^{-4 - 3} b^{1} = 2^{-4} a^{-7} b
$$

Now multiply by $ 3a^{-3} $:

$$
3a^{-3} \cdot 2^{-4} a^{-7} b = 3 \cdot 2^{-4} \cdot a^{-10} \cdot b
$$

$ 2^{-4} = \frac{1}{16} $, so:

$$
= \frac{3}{16} a^{-10} b = \frac{3b}{16 a^{10}}
$$

✔ Answer: $ \boxed{\frac{3b}{16 a^{10}}} $

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

Simplify inside:

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

Now take inverse:

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

✔ Answer: $ \boxed{\frac{3}{2 m^2 n^4}} $

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

Simplify:

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

Now convert:

$$
= \frac{2 p^4 r}{3 q^5}
$$

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

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

Simplify inside:

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

Now invert:

$$
= \frac{4}{3} x^{1} y^{-4} = \frac{4x}{3 y^4}
$$

✔ Answer: $ \boxed{\frac{4x}{3 y^4}} $

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12) $ (3z^{-2} w^3)^2 (2w^{-3})^3 (1w^{-1})^{-2} $

Compute each term:

- $ (3z^{-2} w^3)^2 = 9 z^{-4} w^6 $
- $ (2w^{-3})^3 = 8 w^{-9} $
- $ (1w^{-1})^{-2} = (w^{-1})^{-2} = w^{2} $

Now multiply all:

$$
9 z^{-4} w^6 \cdot 8 w^{-9} \cdot w^2 = 72 z^{-4} w^{6 - 9 + 2} = 72 z^{-4} w^{-1}
$$

Convert to positive exponents:

$$
= \frac{72}{z^4 w}
$$

✔ Answer: $ \boxed{\frac{72}{z^4 w}} $

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13) $ (4a^2 z^{-3})^{-2} (a^{-2} z^3)^{-2} (a^{-1} w^{-2})^{-1} $

Break it down:

1. $ (4a^2 z^{-3})^{-2} = 4^{-2} a^{-4} z^{6} = \frac{1}{16} a^{-4} z^6 $
2. $ (a^{-2} z^3)^{-2} = a^{4} z^{-6} $
3. $ (a^{-1} w^{-2})^{-1} = a^{1} w^{2} $

Now multiply:

$$
\frac{1}{16} a^{-4} z^6 \cdot a^4 z^{-6} \cdot a w^2
$$

Combine:

- $ a^{-4+4+1} = a^1 $
- $ z^{6-6} = z^0 = 1 $
- $ w^2 $
- Constant: $ \frac{1}{16} $

Result: $ \frac{1}{16} a w^2 $

✔ Answer: $ \boxed{\frac{a w^2}{16}} $

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14) $ \left( \frac{8x^{-1} y^2}{2xy^{-3}} \right)^{-1} $

Simplify inside:

$$
\frac{8x^{-1} y^2}{2x y^{-3}} = \frac{8}{2} x^{-1 - 1} y^{2 - (-3)} = 4 x^{-2} y^5
$$

Now invert:

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

✔ Answer: $ \boxed{\frac{x^2}{4 y^5}} $

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

Simplify numerator:

- $ (2z^{-2})^3 = 8 z^{-6} $
- $ (4ab^{-2})^{-1} = \frac{1}{4} a^{-1} b^{2} $

So numerator:

$$
8 z^{-6} \cdot \frac{1}{4} a^{-1} b^2 = 2 z^{-6} a^{-1} b^2
$$

Denominator: $ 4 c^{-3} $

So:

$$
\frac{2 z^{-6} a^{-1} b^2}{4 c^{-3}} = \frac{1}{2} z^{-6} a^{-1} b^2 c^3
$$

Convert to positive exponents:

$$
= \frac{b^2 c^3}{2 a z^6}
$$

✔ Answer: $ \boxed{\frac{b^2 c^3}{2 a z^6}} $

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{1}{x^4 y^4} $ |
| 2 | $ a^5 b^7 $ |
| 3 | $ \frac{n^6}{4 m^4} $ |
| 4 | $ 20 p^{10} q $ |
| 5 | $ 4k^2 $ |
| 6 | $ \frac{1}{a^7} $ |
| 7 | $ \frac{29791 y^{30}}{13824 x^6} $ |
| 8 | $ \frac{3b}{16 a^{10}} $ |
| 9 | $ \frac{3}{2 m^2 n^4} $ |
| 10 | $ \frac{2 p^4 r}{3 q^5} $ |
| 11 | $ \frac{4x}{3 y^4} $ |
| 12 | $ \frac{72}{z^4 w} $ |
| 13 | $ \frac{a w^2}{16} $ |
| 14 | $ \frac{x^2}{4 y^5} $ |
| 15 | $ \frac{b^2 c^3}{2 a z^6} $ |

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