Worksheet for practicing exponent rules with 15 problems to simplify using positive exponents.

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^{-3}}{x^{-2} y} \right)^{-2} $

Step 1: Simplify inside the parentheses:
$$
\frac{x^4 y^{-3}}{x^{-2} y} = x^{4 - (-2)} y^{-3 - 1} = x^{6} y^{-4}
$$

Step 2: Apply the outer exponent $-2$:
$$
(x^6 y^{-4})^{-2} = x^{6 \cdot (-2)} y^{-4 \cdot (-2)} = x^{-12} y^8
$$

Step 3: Write with positive exponents:
$$
\frac{y^8}{x^{12}}
$$

✔ Answer: $ \boxed{\frac{y^8}{x^{12}}} $

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

Step 1: Expand $ (ab^3)^2 = a^2 b^6 $

Step 2: Multiply:
$$
a^3 b^2 \cdot a^2 b^6 = a^{3+2} b^{2+6} = a^5 b^8
$$

✔ Answer: $ \boxed{a^5 b^8} $

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

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

Step 2: Apply exponent $-3$:
$$
(4 m^2 n^2)^{-3} = 4^{-3} m^{-6} n^{-6} = \frac{1}{64} m^{-6} n^{-6}
$$

Step 3: Positive exponents:
$$
\frac{1}{64 m^6 n^6}
$$

✔ Answer: $ \boxed{\frac{1}{64 m^6 n^6}} $

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

Step 1: Expand $ (2p^3 q)^2 = 4 p^6 q^2 $

Step 2: Multiply:
$$
5p^3 q^{-2} \cdot 4p^6 q^2 = 20 p^{3+6} q^{-2+2} = 20 p^9 q^0 = 20 p^9
$$

✔ Answer: $ \boxed{20p^9} $

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5) $ \frac{(4b^{-3})^2 (2b^3)}{4b^{-1}} $

Step 1: Expand numerator:
$$
(4b^{-3})^2 = 16 b^{-6}, \quad \text{so numerator: } 16 b^{-6} \cdot 2b^3 = 32 b^{-3}
$$

Step 2: Divide by denominator:
$$
\frac{32 b^{-3}}{4 b^{-1}} = 8 b^{-3 - (-1)} = 8 b^{-2}
$$

Step 3: Positive exponent:
$$
\frac{8}{b^2}
$$

✔ Answer: $ \boxed{\frac{8}{b^2}} $

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

Step 1: Simplify each term:
- $ (a^{-1})^{-2} = a^{(-1)(-2)} = a^2 $
- $ (a^3)^{-1} = a^{-3} $
- Then multiply: $ a^2 \cdot a^{-3} \cdot a^{-3} = a^{2 - 3 - 3} = a^{-4} $

Step 2: Positive exponent:
$$
\frac{1}{a^4}
$$

✔ Answer: $ \boxed{\frac{1}{a^4}} $

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7) $ \left( \frac{42m^2}{31 m^6} \right)^{-5} $

Step 1: Simplify inside:
$$
\frac{42}{31} m^{2-6} = \frac{42}{31} m^{-4}
$$

Step 2: Apply exponent $-5$:
$$
\left( \frac{42}{31} m^{-4} \right)^{-5} = \left( \frac{42}{31} \right)^{-5} m^{20} = \left( \frac{31}{42} \right)^5 m^{20}
$$

✔ Answer: $ \boxed{\left( \frac{31}{42} \right)^5 m^{20}} $

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

Step 1: Simplify first fraction:
$$
\frac{2^{-1} a^{-4}}{a^3 b^{-2}} = 2^{-1} a^{-4 - 3} b^{2} = \frac{1}{2} a^{-7} b^2
$$

Step 2: Multiply by $ 3a^{-3} $:
$$
\frac{1}{2} a^{-7} b^2 \cdot 3 a^{-3} = \frac{3}{2} a^{-10} b^2
$$

Step 3: Positive exponents:
$$
\frac{3b^2}{2a^{10}}
$$

✔ Answer: $ \boxed{\frac{3b^2}{2a^{10}}} $

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

Step 1: Simplify inside:
$$
\frac{a^{-4} b^{-2}}{a^{-1} b^{-3}} = a^{-4 - (-1)} b^{-2 - (-3)} = a^{-3} b^{1}
$$

Step 2: Apply exponent $-4$:
$$
(a^{-3} b)^{-4} = a^{12} b^{-4}
$$

Step 3: Positive exponents:
$$
\frac{a^{12}}{b^4}
$$

✔ Answer: $ \boxed{\frac{a^{12}}{b^4}} $

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10) $ \frac{8p^3 q^{-3}}{4p^{-2} q^{-2} r^{-1}} $

Step 1: Simplify coefficients and variables:
$$
\frac{8}{4} = 2
$$
$$
p^{3 - (-2)} = p^5, \quad q^{-3 - (-2)} = q^{-1}, \quad r^{0 - (-1)} = r^1
$$

So:
$$
2 p^5 q^{-1} r = \frac{2 p^5 r}{q}
$$

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

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

Step 1: Simplify inside:
$$
\frac{3x^{-2} y^2}{4x^{-4} y^{-3}} = \frac{3}{4} x^{-2 - (-4)} y^{2 - (-3)} = \frac{3}{4} x^2 y^5
$$

Step 2: Apply exponent $-1$:
$$
\left( \frac{3}{4} x^2 y^5 \right)^{-1} = \frac{4}{3} x^{-2} y^{-5} = \frac{4}{3 x^2 y^5}
$$

✔ Answer: $ \boxed{\frac{4}{3x^2 y^5}} $

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12) $ (3x^2 a^3)^2 (2a x^{-3})^2 (16x^2)^{-1} $

Step 1: Expand each part:
- $ (3x^2 a^3)^2 = 9 x^4 a^6 $
- $ (2a x^{-3})^2 = 4 a^2 x^{-6} $
- $ (16x^2)^{-1} = 16^{-1} x^{-2} = \frac{1}{16} x^{-2} $

Step 2: Multiply all together:
$$
9 x^4 a^6 \cdot 4 a^2 x^{-6} \cdot \frac{1}{16} x^{-2}
= (9 \cdot 4 \cdot \frac{1}{16}) a^{6+2} x^{4 - 6 - 2}
= \frac{36}{16} a^8 x^{-4}
= \frac{9}{4} a^8 x^{-4}
$$

Step 3: Positive exponents:
$$
\frac{9 a^8}{4 x^4}
$$

✔ Answer: $ \boxed{\frac{9 a^8}{4 x^4}} $

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

Step 1: Use $ (ab)^{-1} = a^{-1} b^{-1} $, so:

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

Step 2: Multiply all:
$$
\frac{1}{4} a^{-2} x^3 \cdot a^2 x^{-3} \cdot a x^3
= \frac{1}{4} a^{-2 + 2 + 1} x^{3 - 3 + 3} = \frac{1}{4} a^1 x^3
$$

✔ Answer: $ \boxed{\frac{a x^3}{4}} $

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

Step 1: Simplify inside:
$$
\frac{6x^{-3} y^4}{2xy^{-2}} = 3 x^{-3 - 1} y^{4 - (-2)} = 3 x^{-4} y^6
$$

Step 2: Apply exponent $-2$:
$$
(3 x^{-4} y^6)^{-2} = 3^{-2} x^{8} y^{-12} = \frac{1}{9} x^8 y^{-12}
$$

Step 3: Positive exponents:
$$
\frac{x^8}{9 y^{12}}
$$

✔ Answer: $ \boxed{\frac{x^8}{9 y^{12}}} $

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

Step 1: Expand numerator:
- $ (2x^{-2})^3 = 8 x^{-6} $
- $ 4x^3 y^{-1} $
- So total numerator: $ 8 x^{-6} \cdot 4x^3 y^{-1} = 32 x^{-3} y^{-1} $

Step 2: Divide by denominator:
$$
\frac{32 x^{-3} y^{-1}}{4 x^{-1}} = 8 x^{-3 - (-1)} y^{-1} = 8 x^{-2} y^{-1}
$$

Step 3: Positive exponents:
$$
\frac{8}{x^2 y}
$$

✔ Answer: $ \boxed{\frac{8}{x^2 y}} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{y^8}{x^{12}} $ |
| 2 | $ a^5 b^8 $ |
| 3 | $ \frac{1}{64 m^6 n^6} $ |
| 4 | $ 20p^9 $ |
| 5 | $ \frac{8}{b^2} $ |
| 6 | $ \frac{1}{a^4} $ |
| 7 | $ \left( \frac{31}{42} \right)^5 m^{20} $ |
| 8 | $ \frac{3b^2}{2a^{10}} $ |
| 9 | $ \frac{a^{12}}{b^4} $ |
| 10 | $ \frac{2 p^5 r}{q} $ |
| 11 | $ \frac{4}{3x^2 y^5} $ |
| 12 | $ \frac{9 a^8}{4 x^4} $ |
| 13 | $ \frac{a x^3}{4} $ |
| 14 | $ \frac{x^8}{9 y^{12}} $ |
| 15 | $ \frac{8}{x^2 y} $ |

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