# Writing Rational Exponents as Radicals

Want some practice with the other direction?

As long as everything is defined:

$$\cssId{s7}{x^{p/q}} \cssId{s8}{= (x^p)^{1/q}} \cssId{s9}{= \root q\of{x^p}}$$ or $$\cssId{s11}{x^{p/q}} \cssId{s12}{= (x^{1/q})^p} \cssId{s13}{= (\root q\of{x})^p}$$

In both cases, the denominator in the exponent indicates the type of root. The numerator in the exponent is a power, which can go either inside or outside the radical.

## Examples

$x^{1/2} = \sqrt{x}$
$x^{1/3} = \root 3\of {x}$
$\cssId{s21}{x^{3/2}} \cssId{s22}{= \sqrt{x^3}} \cssId{s23}{= (\sqrt{x})^3}$
$\displaystyle x^{-1/2} = \frac{1}{\sqrt{x}}$
$\displaystyle 3x^{-1/5} = \frac{3}{\root 5\of{x}}$
On this exercise, you may assume that $\,x\,$ is positive, so that everything is defined.