Solve Equations With Cube Root 3√x

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


The idea behind solving equations containing cube roots is to raise to power 3 in order to clear the cube root using the property

( 3√x ) 3 = x.

Examples with Solutions

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


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

References and Links

Solve Equations, Systems of Equations and Inequalities.

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