The idea behind solving equations containing square roots is to raise to power 3 in order to clear the cube root using the property
( ^{3}√x )^{ 3} = x.
Example 1 :
Find all real solutions to the equation
^{3}√x  x = 0
Solution to Example 1:

Rewrite equation with the term containing cube root on one side as follows.
^{3}√x = x

Raise both sides to power 3 in order to clear the cube root.
( ^{3}√x )^{ 3} = x ^{ 3}

Rewrite the above equation with right side equal to zero.
x  x ^{ 3} = 0

Factor
x (1  x ^{ 2}) = 0

and solve for x.
solutions are : x = 0 , x =  1 and x = 1.
Check the solutions found.
1. x = 0
Left side (LS) of the given equation when x = 0
LS = ^{3}√x  x = ^{3}√(0)  0 = 0
Right Side (RS) of the given equation when x = 0
RS = 0
2. x = 1
Left side (LS) of the given equation when x = 1
LS = ^{3}√x  x = ^{3}√(1)  (1) = 1 + 1 = 0
Right Side (RS) of the given equation when x = 1
RS = 0
3. x = 1
Left side (LS) of the given equation when x = 1
LS = ^{3}√x  x = ^{3}√(1)  1 = 0
Right Side (RS) of the given equation when x = 1
RS = 0
Example 2 :
Find all real solutions to the equation
^{3}√( x^{ 2} + 2 x + 8 ) = 2
Solution to Example 2:

Given
^{3}√( x^{ 2} + 2 x + 8 ) = 2

We raise both sides to power 3 in order to clear the cube root.
[ ^{3}√( x^{ 2} + 2 x + 8 ) ]^{ 3} = 2 ^{ 3}

and simplify.
x^{ 2} + 2 x + 8 = 8

Rewrite the above equation with right side equal to zero.
x^{ 2} + 2 x = 0

Factor
x (x + 2) = 0

and solve for x.
x = 0 and x =  2.
Let us check the solutions obtained as an exercise.
1. x = 0
Left side (LS) of the given equation when x = 0
LS = ^{3}√( x^{ 2} + 2 x + 8 ) = cube_root (0 + 0 + 8) = 2
Right Side (RS) of the given equation when x = 0
RS = 2
2. x = 2
Left side (LS) of the given equation when x = 0
LS = ^{3}√( x^{ 2} + 2 x + 8 )
= ^{3}√( (2)^{ 2} + 2*(2) + 8 ) = cube_root ( 8 ) = 2
Right Side (RS) of the given equation when x = 0
RS = 2
Exercises:(answers further down the page)
Solve the following equations
1. ^{3}√x  4 x = 0
2. ^{3}√( x^{ 2} + 2 x + 61 ) = 4
Solutions to above exercises
1. x = 0 , x = 1 / 8 , x =  1 / 8
2. x = 1 , x = 3
More References and links
Solve Equations, Systems of Equations and Inequalities.