Solve Equations With Cube Root

Tutorial on how to solve equations containing cube roots. Detailed solutions to examples, explanations and exercises are included.

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.