How do I express the domain interval of this function?

It is called fifth root
(the word "square" is reserved for "second")

Odd-powered roots usually do not have restrictions.

Is there a 5th root for -32?

Yes, because (-2)^5 = -32

Domain problems, for roots, normally occur when you have even-powered roots (such as the square root or the 4th root) because it is impossible to take the even-powered root of a negative number.

What is the 4th root of -16?
It is not -2, because (-2)^4 = +16

So, the fifth root does not restrict your domain.

Next, you look at the argument itself (the binomial of which you take the fifth root), and it is exactly that, a binomial.

Polynomials do not have any restrictions on their domains.

Therefore, there does not seem to be any restrictions on the domain.

The domain can be written in many different ways:

"All the real numbers"

Domain = ℝ
(ℝ is the symbole that represents "all the real numbers")

Domain = (-∞, +∞)
or (-∞, ∞)

Most people prefer to use the interval notiation without putting the + sign in front of the "plus infinity" sign. I do use it, simply to avoid confusion.
Note that the notation always uses round brackets for infinites, as they are not real numbers (they are never included in the domain).

Domain = { x : x ∈ ℝ }

and so on.

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The usual interval notation uses round brackets when the border number is NOT included in the domain, and a square bracket when the number is included.

For example: (-1, 3]
means all the numbers between -1 and 3, including 3 itself, but NOT including -1.

Some books use square brackets only, with the bracket facing in if the number is OK, but facing our if it is not OK.

(-1, 3] becomes ]-1, 3]