Negative Exponents

A negative exponent means one over, just as it did with whole numbers: ${5}^{-2} = \frac{1}{{5}^{2}} = \frac{1}{25}$. The minus in the exponent does not flip the sign of the answer. It moves the power below the line. One over a fraction turns it upside down, so a fraction base flips: ${\frac{2}{5}}^{-2} = {\frac{5}{2}}^{2} = \frac{25}{4}$.

With a fraction exponent, deal with the minus first, then the fraction as in the last lesson. Take ${25}^{\frac{-3}{2}}$:

$${25}^{\frac{-3}{2}} = \frac{1}{{25}^{\frac{3}{2}}} = \frac{1}{{\left({25}^{\frac{1}{2}}\right)}^{3}} = \frac{1}{{5}^{3}} = \frac{1}{125}$$

Why one over? Multiply ${25}^{\frac{3}{2}}$ by ${25}^{\frac{-3}{2}}$ and the exponents add to 0, and ${25}^{0} = 1$. Two numbers whose product is 1 are each other's reciprocals, so ${25}^{\frac{-3}{2}}$ must be one over ${25}^{\frac{3}{2}}$.