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^(3/2) = 64. Raise both sides to the power 2/3: x is 64^(2/3). The cube root of 64 is 4, and 4 squared is 16, so x is 16.

Check it: 16^(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^(3/2) is (√(x))^3, and undoing a square root means squaring, which hides a sign. Take x^(1/2) = -6: squaring both sides gives x = 36, but 36^(1/2) is 6, not -6. So 36 is a false answer, and that equation has none.