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Solving Radical Equations:
     Higher-Index Examples
(page 6 of 6)

  • Solve the equation:  cbrt(2x – 5) = 3
  • Since this is a CUBE root, rather than a square root, I undo the radical by cubing both sides of the equations, rather than squaring:


    x = 16


    I must remember to check my solution:


    cbrt(27) =? 3

                   3 = 3   ...yes!

    So the solution is x = 16. Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

  • Solve the equation:  cbrt(x^3 – 3x^2) + 1 = x
  • (Note that the "plus one" is outside the cube root.)

    Since this is a cube root, I'll cube both sides to undo the radical. But first, I want to isolate the radical:


    1/3 = x


    Remember to check the solution:


    1/3 = 1/3

    So the solution is x = 1/3.

  • Solve the equation:  x + 1 = (x^4 + 4x^3 – x)^(1/4)
  • Since this is a fourth root, I'll raise both sides to the fourth power:


    x = –1/2, x = –1/3


    Then I'll check my answers:


    x = –1/2:

    1/2 = 1/2


    I'll leave the other check for you. However, the graph does indicate that both solutions are valid.


    Graphing the left- and right-hand sides of the original equation:

    y = x + 1; y = (x^4 + 4x^3 – x)^(1/4) get the picture at right:






    Zooming in, you can see that the lines seem to intersect...


    zoomed in a bit




    ...and, zooming in some more, you can see the two solutions:


    graph of solutions



    Remember that you can't have negatives inside a fourth root. That's why the green line is broken into pieces like that: you can only graph where x4 + 4x3x is non-negative, which occurs in three pieces, where the graph is at or above the x-axis.


    y = x^4 + 4x^3 – x

    Then the solution is  x = – 1/2, – 1/3.

Since cube roots can have negative numbers inside them, you don't tend to have the difficulty with them regarding checking the answers that you did with square roots. However, you will have those difficulties with fourth roots, sixth roots, eighth roots, etc; namely, any even-index root. Be careful!

You may or may not be required to show solutions graphically, but if you have a graphing calculator (so drawing the graphs is just a matter of quickly punching a few buttons), you can use the graphs to check your work on tests. In any case, be careful with your squaring ("Square sides, not terms!"), do each step carefully, and don't forget to "Check your solutions!"

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Cite this article as:

Stapel, Elizabeth. "Solving Radical Equations: Higher-Index Examples." Purplemath. Available from Accessed



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