When a fraction is raised to a negative exponent, do you normally transform it to 1 over the fraction, or invert the fraction?

Question

My text shows that

$$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$

It also shows that

$$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$

In the first equation, it seems that $(3a^2/4b)^{-3}$ was inverted to $1/(3a^2/4b)$ with the denominator raised to the power of $3$: $1/(3a^2/4b)^3$.

In the second equation, $1/(144/b)$ is equal to $(144/b)^{-1}$ (correct?), and if I do what was done in the first equation, I get what I started with: $1/(144/b)$.

So how is it that I convert $1/(144/b)$ to $b/144$? How does that accord with what happened in the first equation?

Thank you.

-Hal

Answer 1

Each method you suggest leads to the equivalent result. $$\left(\frac{144}{b}\right)^{-1} = \frac 1{\frac {144}b} \cdot \frac bb = \frac b{144}$$

Answer 2

You forgot to negate the exponent $-1.\;$If you follow the recipe you get $$\left(\frac{144}{b}\right)^{-1} = \left(\frac{b}{144}\right)^{1} = \frac{b}{144} $$

Answer 3

These transformations between different forms of the expression are not rules about what you must do every time you see a denominator that is a fraction or when you see a negative power. They are merely different ways of describing or computing the same quantity. You may sometimes use these facts to rewrite an expression for some quantity when you find that the way it is initially expressed is inconvenient for some reason.

So, both facts are true and give you ways to rewrite expressions without changing their values, but the question is how to know when they will be useful to you. That is something that is hard to describe in a simple way that covers all possible situations you may encounter, but it is something that becomes easier as you gain more experience.