Return to the Purplemath home page

 The Purplemath Forums
Helping students gain understanding
and self-confidence in algebra


powered by FreeFind

 

Return to the Lessons Index  | Do the Lessons in Order  |  Get "Purplemath on CD" for offline use  |  Print-friendly page



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)

    ...you get the picture at right:

      

      

    graph
       

      

      

    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 + 4x3 x 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!"

<< Previous  Top  |  1 | 2 | 3 | 4 | 5 | 6  |  Return to Index

Cite this article as:

Stapel, Elizabeth. "Solving Radical Equations: Higher-Index Examples." Purplemath. Available from
    http://www.purplemath.com/modules/solverad6.htm. Accessed
 

 



Purplemath:
  Linking to this site
  Printing pages
  School licensing


Reviews of
Internet Sites:
   Free Help
   Practice
   Et Cetera

The "Homework
   Guidelines"

Study Skills Survey

Tutoring from Purplemath
Find a local math tutor


This lesson may be printed out for your personal use.

Content copyright protected by Copyscape website plagiarism search

  Copyright 2002-2012  Elizabeth Stapel   |   About   |   Terms of Use

 

 Feedback   |   Error?