Negative exponents
A negative exponent does not make a number negative. It means “take the reciprocal.” The key rule is
- (a^{-n} = \dfrac{1}{a^n}), for (a \neq 0)
So the exponent becomes positive after you move the base across the fraction bar.
How to simplify
- Identify the negative exponent.
- If it is in the numerator, move the factor to the denominator.
- If it is in the denominator, move it to the numerator.
- Change the exponent to positive.
- Simplify any powers. Compute the positive exponent if possible.
- Reduce the fraction. Cancel common factors only when they are multiplied, not added.
Examples of the idea
- (x^{-3} = \dfrac{1}{x^3})
- (\dfrac{1}{y^{-2}} = y^2)
- (\dfrac{a^{-1}b^2}{c^{-3}} = \dfrac{b^2c^3}{a})
Quick check
Your final answer should have no negative exponents. If one remains, rewrite the expression again using the reciprocal rule. Also check that you did not change the value by accidentally flipping terms that were being added or subtracted. The result should be fully simplified and exact.