Simplify or otherwise reduce (4/9)^-3/2

[JoeK882]

(As directed, I am reposting this question in the correct section)

This is question 4 in Chapter 5 of MGMAT Number Properties guide on page 71.

Simplify or otherwise reduce the following expressions using the rules of exponents: (4/9)^-3/2

Here are the steps in the answer key:

(4/9)^-3/2 = (9/4)^3/2 (I get this) = ±√9/4^3 (I get this) = (±√9/4^3) (not sure why this happens) = (3/2)^3 (not sure) =3^3/2^3 =27/8

If someone can break down the steps in basic, simple english that would be great. Thanks!

[RonPurewal]

The easiest way to apprehend fractional and/or negative exponents is to think of them as instructions.

The exponent has up to three components:

- a numerator (top of fraction)
This is a normal exponent. It raises stuff to a power.

"- a denominator (bottom of fraction)
This is a root.
If it's a 2, it's a square root.
If it's a 3, it's a cube root.
And so on.

"- a negative sign
This makes a reciprocal (= flips the number around).

NONE of these are ambiguous. There should be no "±" signs anywhere. (Symbols don't mean more than one thing. If they could, they would be useless.)

You can execute these steps in any order whatsoever. Generally, it's easier to do the root before the power, just so you're working with smaller numbers. But any order will give the same result.

So...
4/9 to the power -3/2

means...
flip
cube
square root
In whatever order. You probably want to do the square root before the cube (unless evaluating things like √729 is your idea of fun).
If you start with 4/9 and do these steps to it, you'll end up with 27/8.