The idea behind solving equations containing square roots is to raise to power 3 in order to clear the cube root using the property
( cube_root( x ) )^{ 3} = x.
Example 1 : Find all real solutions to the equation
cube_root( x )  x = 0
Solution to Example 1:
 Rewrite equation with the term containing cube root isolated
cube_root( x ) = x
 Raise both sides to power 3 in order to clear the cube root.
[ cube_root( 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.
It is good to check the solutions found.
1. x = 0
Left side (LS) of the given equation when x = 0
LS = cube_root( x )  x = cube_root (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 = cube_root( x )  x = cube_root (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 = cube_root( x )  x = cube_root (1)  1 = 0
Right Side (RS) of the given equation when x = 1
RS = 0
Example 2 : Find all real solutions to the equation
cube_root( x^{ 2} + 2 x + 8 ) = 2
Solution to Example 2:
 Given
cube_root( x^{ 2} + 2 x + 8 ) = 2
 We raise both sides to power 3 in order to clear the cube root.
[ cube_root( 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 = cube_root( 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 = cube_root( x^{ 2} + 2 x + 8 )
= cube_root ( (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. cube_root( x )  4 x = 0
2. cube_root( 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 on how to Solve Equations, Systems of Equations and Inequalities.
