Answer
$-\dfrac{1}{27}$
Work Step by Step
Use the rule $a^{-m}=\left(\dfrac{1}{a}\right)^m$:
$-81^{-\frac{3}{4}} =-\left (\dfrac{1}{81}\right )^{\frac{3}{4}} $
Use $81=3^4$.
$=-\left (\dfrac{1}{3^4}\right )^{\frac{3}{4}} $
Use the rule $\left(\dfrac{a}{b}\right)^n=\dfrac{a^m}{b^m}$:
$=-\dfrac{1^{\frac{3}{4}}}{(3^4)^{\frac{3}{4}}} $
Use the rule $\left(a^m\right)^n=a^{mn}$, then simplify to obtain:
$=-\dfrac{1}{3^{3}}$
$=-\dfrac{1}{27}$
Hence, the correct answer is $-\frac{1}{27}$.
