Practice converting negative exponents to positive exponents with this math worksheet.

Let's solve each of these problems step by step, using the rule for negative exponents:

> Rule: $ a^{-n} = \frac{1}{a^n} $
> Or conversely: $ \frac{1}{a^{-n}} = a^n $

We will rewrite each expression with positive exponents only, as instructed.

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1. $ 5^{-1} $

Flip the base:
$$
5^{-1} = \frac{1}{5}
$$

✔ Answer: $ \frac{1}{5} $

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2. $ 4^{-9} $

Flip the base:
$$
4^{-9} = \frac{1}{4^9}
$$

✔ Answer: $ \frac{1}{4^9} $

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3. $ \frac{1}{3^{-2}} $

Here, the negative exponent is in the denominator. Flip it to move it to the numerator:
$$
\frac{1}{3^{-2}} = 3^2 = 9
$$

✔ Answer: $ 9 $

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4. $ \frac{1}{y^{-2}} $

Same logic: flip the base:
$$
\frac{1}{y^{-2}} = y^2
$$

✔ Answer: $ y^2 $

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5. $ v^{-4} $

Flip the base:
$$
v^{-4} = \frac{1}{v^4}
$$

✔ Answer: $ \frac{1}{v^4} $

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6. $ a b^{-2} $

Only $ b $ has a negative exponent. Keep $ a $, and flip $ b^{-2} $:
$$
a b^{-2} = a \cdot \frac{1}{b^2} = \frac{a}{b^2}
$$

✔ Answer: $ \frac{a}{b^2} $

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7. $ \frac{1}{3x^{-3}} $

Break this down: $ x^{-3} $ is in the denominator. So we can write:
$$
\frac{1}{3x^{-3}} = \frac{1}{3} \cdot \frac{1}{x^{-3}} = \frac{1}{3} \cdot x^3 = \frac{x^3}{3}
$$

✔ Answer: $ \frac{x^3}{3} $

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8. $ \frac{1}{a^2 b^{-3}} $

We have $ b^{-3} $ in the denominator. Flip that:
$$
\frac{1}{a^2 b^{-3}} = \frac{1}{a^2} \cdot \frac{1}{b^{-3}} = \frac{1}{a^2} \cdot b^3 = \frac{b^3}{a^2}
$$

✔ Answer: $ \frac{b^3}{a^2} $

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9. $ \frac{-3}{x^{-2} y^{-3}} $

Negative exponents in the denominator → flip them to numerator.

First, note that $ -3 $ is just a constant (negative sign), so keep it.

So:
$$
\frac{-3}{x^{-2} y^{-3}} = -3 \cdot \frac{1}{x^{-2}} \cdot \frac{1}{y^{-3}} = -3 \cdot x^2 \cdot y^3
$$

✔ Answer: $ -3x^2 y^3 $

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10. $ \frac{p^2 q^{-1}}{a^4 b^{-3}} $

We have two negative exponents: $ q^{-1} $ in numerator and $ b^{-3} $ in denominator.

- $ q^{-1} $ → $ \frac{1}{q} $
- $ b^{-3} $ in denominator → becomes $ b^3 $ in numerator

So:
$$
\frac{p^2 q^{-1}}{a^4 b^{-3}} = \frac{p^2}{q} \cdot \frac{b^3}{a^4} = \frac{p^2 b^3}{a^4 q}
$$

✔ Answer: $ \frac{p^2 b^3}{a^4 q} $

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✔ Final Answers (with positive exponents):

1. $ \frac{1}{5} $
2. $ \frac{1}{4^9} $
3. $ 9 $
4. $ y^2 $
5. $ \frac{1}{v^4} $
6. $ \frac{a}{b^2} $
7. $ \frac{x^3}{3} $
8. $ \frac{b^3}{a^2} $
9. $ -3x^2 y^3 $
10. $ \frac{p^2 b^3}{a^4 q} $

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Let me know if you'd like a visual explanation or more practice!