(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:
Remember to check the solution:
So the solution is x = 1/3.
Solve the equation:
Since this is a fourth root, I'll raise
both sides to the fourth power:
Then I'll check my answers:
x = –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:
...you get the picture at right:
Zooming in, you can see that the lines seem
...and, zooming in some more, you can see
the two 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.
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!"