Fractional Powers

For x not negative, a fractional power is undone by its reciprocal power, because powers of powers multiply: 3/2 times 2/3 is 1. Take ${x}^{\frac{3}{2}} = 64$. Raise both sides to the power 2/3: x is ${64}^{\frac{2}{3}}$. The cube root of 64 is 4, and 4 squared is 16, so x is 16.

Check it: ${16}^{\frac{3}{2}}$ is the square root of 16, cubed, which is 4 cubed, 64. The check matters because the bottom of 3/2 is even. ${x}^{\frac{3}{2}}$ is ${\left(\sqrt{x}\right)}^{3}$, and undoing a square root means squaring, which hides a sign. Take ${x}^{\frac{1}{2}} = -6$: squaring both sides gives x = 36, but ${36}^{\frac{1}{2}}$ is 6, not -6. So 36 is a false answer, and that equation has none.