Answer
$\color{blue}{\dfrac{1000}{1331}}$
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:
$\left(\dfrac{121}{100}\right)^{-3/2} = \dfrac{1}{(\frac{121}{100})^{3/2}}$
Use rule (1) above to obtain:
$=\dfrac{1}{[(\frac{121}{100})^{1/2}]^3}$
Use rule (2) above to obtain:
$=\dfrac{1}{\left(\sqrt[2]{\frac{121}{100}}\right)^3}$
With $\dfrac{121}{100}=\left(\dfrac{11}{10}\right)^2$, the expression above is equivalent to:
$=\dfrac{1}{\left(\sqrt[2]{(\frac{11}{10})^2}\right)^3}$
Use rule (3) above to obtain:
$=\dfrac{1}{\left(\frac{11}{10}\right)^3}$
Use rule (6) above to obtain:
$\\=\dfrac{1}{\frac{11^3}{10^3}}
\\=\dfrac{1}{\frac{1331}{1000}}
\\=1 \cdot \dfrac{1000}{1331}
\\=\color{blue}{\dfrac{1000}{1331}}$
