One Over n

So far exponents have been whole numbers. To give ${49}^{\frac{1}{2}}$ a meaning, keep the rule for a power of a power, which multiplies the exponents: ${\left({49}^{\frac{1}{2}}\right)}^{2} = {49}^{1}$. So ${49}^{\frac{1}{2}}$ is a number whose square is 49, and like the root sign it means the non-negative one: $\sqrt{49}$, which is 7.

The same works for any n: ${a}^{\frac{1}{n}}$ is the nth root of a, the number whose nth power is a. For an even n, a must not be negative, and the root is the non-negative one. A power of 1/3 is a cube root, ${125}^{\frac{1}{3}} = cbrt\left(125\right) = 5$, because ${5}^{3} = 125$. A power of 1/4 is a fourth root, ${10000}^{\frac{1}{4}} = 10$, because ${10}^{4} = 10000$.

Why write roots as powers at all? Because then every exponent rule you already know works on roots too, and the rest of this chapter uses exactly that. The bottom of the fraction is the root's index: 2 for a square root, 3 for a cube root.