Answer
No, $\frac{\frac{a}{b}}{c} \neq \frac{a}{\frac{b}{c}}$ in most cases.
When c = 1 or -1, then the equality holds.
Work Step by Step
We first write $\frac{\frac{a}{b}}{c}$ as a quotient:
$\frac{\frac{a}{b}}{c} = \frac{a}{b}\div{c}$
Note that the reciprocal of $\frac{x}{y}$ is $\frac{y}{x}$. Also, dividing by a fraction is the same as multiplying by that fraction's reciprocal.
$\frac{a}{b}\cdot\frac{1}{c} = \frac{a}{bc}$
Now write $\frac{a}{\frac{b}{c}}$ as a quotient:
$\frac{a}{\frac{b}{c}} = a \div \frac{b}{c} = a \cdot \frac{c}{b} = \frac{ac}{b}$
If the two fractions were to equal, then $\frac{a}{bc} = \frac{ac}{b} \rightarrow \frac{\frac{a}{bc}}{\frac{ac}{b}} = 1 \rightarrow \frac{1}{c^2} = 1 \rightarrow c^2 = 1$
So these two fractions are equal if $c = -1, 1$
