Two Thirds Power

In x^(2/3) the top of the exponent is even: x^(2/3) is (cbrt(x))^2, the cube root of x, squared. A square hides a sign, so two numbers can share this power. Take x^(2/3) = 9. The cube root of 27 is 3, and 3 squared is 9, so 27 works. The cube root of -27 is -3, and (-3)^2 is also 9, so -27 works too.

To solve it, raise both sides to the power 3/2. The right side becomes 27, the positive number whose 2/3 power is 9, as above. The left side does not become x: the power 2/3 squared away the sign of x, and the power 3/2 takes a square root, which cannot bring a sign back. So the left side becomes |x|:

(x^(2/3))^(3/2) = |x|

|x| = 27 means x = 27 or x = -27, the same way x^2 = 9 gives 3 and -3.

Two quick tests come from the same picture. The bottom, 3, is odd, so the cube root takes any number and neither answer has to be thrown out. The top, 2, is even, so the power is never negative: x^(2/3) = -9 has no real answer.