Fractions and Negatives

A negative exponent means a reciprocal: ${3}^{-1} = \frac{1}{3}$. So a fraction like 1/3 is a power of 3, and powers of 1/3 are powers of 3 with the sign of the exponent flipped. For example, $\left(\frac{1}{3}\right)^{-2}$ is ${3}^{2}$, which is 9.

Use that to solve $\left(\frac{1}{3}\right)^{x} = 9$. Write 9 as a power of 1/3: $9 = \left(\frac{1}{3}\right)^{-2}$. The equation reads $\left(\frac{1}{3}\right)^{x} = \left(\frac{1}{3}\right)^{-2}$, the same base on both sides, so x = -2. Check: $\left(\frac{1}{3}\right)^{-2}$ flips to ${3}^{2}$, which is 9.

The negative answer makes sense. Positive powers of 1/3 shrink below 1, so to land on 9, above 1, the exponent has to be negative.