#### Answer

$-\frac{1}{512}$

#### Work Step by Step

First, we will simplify the expression within the parenthesis by finding the least common denominator:
$(\frac{3}{4}-\frac{7}{8})^{3}$
=$(\frac{3(2)-1(7)}{8})^{3}$
=$(\frac{6-7}{8})^{3}$
=$(\frac{-1}{8})^{3}$
Next, we raise both the numerator and the denominator to the power of $3$ and simplify:
$(\frac{-1}{8})^{3}=\frac{(-1)^{3}}{8^{3}}=\frac{-1\times-1\times-1}{8\times8\times8}=\frac{-1}{512}=-\frac{1}{512}$