Answer
$\color{blue}{\dfrac{256}{81}}$
Work Step by Step
RECALL:
(1) $a^{m/n} = \left(a^{1/n}\right)^m$
(2) $a^{1/n} = \sqrt[n]{a}$
(3) For positive real numbers $a$, $\sqrt[n]{a^n}=a$
(4) $a^{-m} = \dfrac{1}{a^m}$
(5) When $n$ is odd, $\sqrt[n]{a^n} = n$
(6) $\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$
Use rule (4) above to obtain:
$\left(\dfrac{27}{64}\right)^{-4/3} = \dfrac{1}{(\frac{27}{64})^{4/3}}$
Use rule (1) above to obtain:
$=\dfrac{1}{[(\frac{27}{64})^{1/3}]^4}$
Use rule (2) above to obtain:
$=\dfrac{1}{\left(\sqrt[3]{\frac{27}{64}}\right)^4}$
With $\dfrac{27}{64}=\left(\dfrac{3}{4}\right)^3$, the expression above is equivalent to:
$=\dfrac{1}{\left(\sqrt[3]{(\frac{3}{4})^3}\right)^4}$
Use rule (5) above to obtain:
$=\dfrac{1}{\left(\frac{3}{4}\right)^4}$
Use rule (6) above to obtain:
$\\=\dfrac{1}{\frac{3^4}{4^4}}
\\=\dfrac{1}{\frac{81}{256}}
\\=1 \cdot \dfrac{256}{81}
\\=\color{blue}{\dfrac{256}{81}}$
