# If #root(3)(3(root(3)x - 1/(root(3)x))) = 2#, then #root(3)x + 1/(root(3)x) =# what?

##### 1 Answer

We start with the original function:

and we want to solve for:

which means we need to do a bit more work and solve for

We then take each side to the exponent 3 (which gets rid of the cube root on the left hand side):

Taking the cube root of a number is equivalent to taking a fractional exponent of 1/3 and so this simplifies to:

We then divide both sides by 3 (to get ride of the multiplication by 3 on the left hand side)

Next we multiply both sides by

You might not immediately recognize it, but this is a quadratic equation. To make that more obvious, we're going to temporarily substitute

And now our equation becomes:

which more closely resembled what we expect from a quadratic equation. Using the quadratic formula and the letter

we find that:

Now we'll substitute back our function of x for y:

and

So we have two solutions for x:

Using either of these values (

We know that