Isolate the cube root:



Raise both sides to the 3rd power (that is, "cube" both sides)



Raising a radical to the same exponent as its index amounts
to removing the radical and the exponent, so we have:



But we must multiply the right side out:









Get 0 on one side:



We like the 0 to be on the right, so we switch sides:



Factor  out of the first two terms on the left:



Factor  out of the last two terms on the left:



Factor  out of both terms on the left:



Factor the second parentheses as the difference of sqwuares:



Use the zero-factor principle and set each factor = 0:

x-3=0 gives solution x=3

x-1=0 gives solution x=1

x+1=0 gives solution x=-1

The solutions are -1, 1, and 3.  They all check in the original.

(You only need to check radical equations for extraneous solutions
when there is a root with an even index or a denominator containg
a variable.)

Edwin