Question

How do you simplify `-4^(-5/2)`?

Answer 1

Let's start with the negative exponent.

We can appy the following power rule: `x^(-1) = 1/x` or, more general: `x^(-a) = 1 / x^a`.

So,
`-4^(-5/2) = - 1/4^(5/2)`.

As next, let's take care of the `5/2`.
First of all, the fraction can be split into `5/2 = 5 * 1/2`.

Another power rule states: `x^(a*b) = (x^a)^b`.
Here, this means that you can either compute `(4^5)^(1/2)` or `(4^(1/2))^5` instead of `4^(5/2)`.

Let's stick with `(4^(1/2))^5`.

What does `4^(1/2)` mean? `x^(1/2) = sqrt(x)`, so `4^(1/2) = sqrt(4) = 2`.
So you get `4^(5/2) = (4^(1/2))^5 = (4^(1/2))^5 = (sqrt(4))^5 = 2^5 = 32`.

In total, your solution is:

`-4^(-5/2) = - 1/4^(5/2) = - 1 / ((4^(1/2))^5) = - 1/((sqrt(4))^5) = - 1 / 2^5 = - 1 / 32`.

Answer 2

• Make the exponent positive
• A bit of algebra

Explanation:

Let's start off by having a look at the expression. The minus sign is not a part of the base (which is 4), so therefore, we can write the expression like this:
`-(4^(-5/2))`
When we have to deal with negative exponents, we have a set of rules that simplyfies them. One of them is:
`a^(-n) = 1/(a^n)`
Another one, that we will use to convert exponents into square roots is:
`a^(m/n) = root(n)a^m`

So let's begin!
We already have `-(4^(-5/2))`, so we will just star by using the first rule (the one about making exponents positive).

`-(4^(-5/2)) = -1/(4^(5/2))`

Followed by the rule about exponents into square roots:

`-1/4^(5/2) = -1/(root(2)4^5) = -1/(sqrt(4^5)`

Just some algebra that remains!

`-1/(sqrt(4^5)) = -1/(2^5) = -1/32`