(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 to intersect...

...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
x^{4}
+ 4x^{3} – 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!"