Answer
$\dfrac{125a^{15}}{8b^{6}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\left( \dfrac{2}{5}a^{-5}b^{2} \right)^{-3}
,$ use the laws of exponents.
$\bf{\text{Solution Details:}}$
Using the extended Power Rule of the laws of exponents, which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( \dfrac{2}{5}a^{-5}b^{2} \right)^{-3}
\\\\=
\left( \dfrac{2}{5}\right)^{-3}a^{-5(-3)}b^{2(-3)}
\\\\=
\left( \dfrac{2}{5}\right)^{-3}a^{15}b^{-6}
.\end{array}
Using the Power of a Quotient Rule of the laws of exponents, which is given by $\left( \dfrac{x^m}{y^n} \right)^p=\dfrac{x^{mp}}{y^{np}},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( \dfrac{2}{5}\right)^{-3}a^{15}b^{-6}
\\\\=
\dfrac{2^{-3}}{5^{-3}}a^{15}b^{-6}
\\\\=
\dfrac{2^{-3}a^{15}b^{-6}}{5^{-3}}
.\end{array}
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2^{-3}a^{15}b^{-6}}{5^{-3}}
\\\\=
\dfrac{5^{3}a^{15}}{2^{3}b^{6}}
\\\\=
\dfrac{125a^{15}}{8b^{6}}
.\end{array}
