Tutoring high school math, radicals are prominent.  The tutor offers an example of simplifying a higher radical.

I’ve written a number of articles on simplifying radicals. I won’t encumber this one with a list of the previous ones, but you can find them by searching for “radical” or “root” in the search box.

Today, we’ll look at this example. (Note that the fifth root is the same as the exponent 1/5.)

Simplify (-486)^(1/5)

Note that you can simplify odd roots of negatives, just not even ones.

If you try to take the fifth root of -486 on your calculator, you’ll get a messy decimal, which is not acceptable as simplified form. Therefore, a perfect fifth power number must lurk inside -486; the job is to find it.

We start by listing perfect fifth powers:

2^5=32
3^5=243
4^5=1024

Clearly, 4^5 is much too large to fit in -486; therefore, our number must be either 32 or 243. 32 looks tempting, by if you try -486/32 you get a decimal. However,

-486/243=-2

Therefore,

(-486)^(1/5) = (-243)^(1/5)*(2)^(1/5) = -3*(2)^(1/5)

When simplifying an odd root of a negative, always take the negative out front:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.