# Why are fractional exponents roots?

Mar 23, 2015

One of the basic observations about integer exponents can be expressed as:
${a}^{p} \times {a}^{q} = {a}^{p + q}$
${a}^{p} \times {a}^{q} \times {a}^{r} = {a}^{p + q + r}$
and so on (post another question if it isn't clear why this is true for integers)

If this observation is extended to fractional exponents we would have, for example, something like:
${a}^{\frac{1}{3}} \times {a}^{\frac{1}{3}} \times {a}^{\frac{1}{3}} = {a}^{\frac{1}{3} + \frac{1}{3} + \frac{1}{3}} = {a}^{1} = a$

If three identical values multiplied together equal $a$
then the values must be $\sqrt[3]{a}$