Exponent Rules Still Work

The rules you learned for whole-number exponents hold for fractions too, as long as the base is positive. Multiplying powers of the same base adds the exponents: ${x}^{\frac{1}{3}}\cdot {x}^{\frac{2}{3}} = {x}^{\frac{1}{3}+\frac{2}{3}} = {x}^{1}$, which is just x.

This agrees with what the powers mean. ${x}^{\frac{1}{3}}$ is the cube root of x, so three of them multiplied should give x back: ${x}^{\frac{1}{3}}\cdot {x}^{\frac{1}{3}}\cdot {x}^{\frac{1}{3}} = {x}^{\frac{3}{3}}$, which is x. In the same way $\sqrt{x}\cdot \sqrt{x} = {x}^{\frac{1}{2}+\frac{1}{2}}$, which is x.

The other rules carry over the same way. Dividing subtracts exponents, $\frac{{x}^{\frac{5}{4}}}{{x}^{\frac{1}{4}}} = {x}^{\frac{4}{4}}$, and a power of a power multiplies them, ${\left({x}^{\frac{1}{3}}\right)}^{9} = {x}^{3}$. Add or subtract the fractions over a common denominator, then simplify: an exponent of 4/4 is 1.