We need to simplify the following expressions: a) $(\frac{5}{2})^{-1} \frac{5^{-1}}{2^{-1}}$ b) $(\frac{5}{2x})^{-1}$ c) $(\frac{5x}{2})^{-1}$

1. Problem Description

We need to simplify the following expressions:

a) $(\frac{5}{2})^{-1} \frac{5^{-1}}{2^{-1}}$

b) $(\frac{5}{2x})^{-1}$

c) $(\frac{5x}{2})^{-1}$

2. Solution Steps

a) $(\frac{5}{2})^{-1} \frac{5^{-1}}{2^{-1}}$

Recall that $(a/b)^{-1} = b/a$ and $a^{-1} = \frac{1}{a}$.

Then

$(\frac{5}{2})^{-1} = \frac{2}{5}$.

Also,

$\frac{5^{-1}}{2^{-1}} = \frac{\frac{1}{5}}{\frac{1}{2}} = \frac{1}{5} \cdot \frac{2}{1} = \frac{2}{5}$.

Therefore,

$(\frac{5}{2})^{-1} \frac{5^{-1}}{2^{-1}} = \frac{2}{5} \cdot \frac{2}{5} = \frac{4}{25}$.

b) $(\frac{5}{2x})^{-1}$

Using the formula $(a/b)^{-1} = b/a$, we have:

$(\frac{5}{2x})^{-1} = \frac{2x}{5}$.

c) $(\frac{5x}{2})^{-1}$

Using the formula $(a/b)^{-1} = b/a$, we have:

$(\frac{5x}{2})^{-1} = \frac{2}{5x}$.

3. Final Answer

a) $\frac{4}{25}$

b) $\frac{2x}{5}$

c) $\frac{2}{5x}$