Answer
$-15\sqrt[3]{-75}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[3]{25(-3)^4(5)^3}
,$ use the laws of exponents to simplify the radicand. Then, find a factor of the radicand that is a perfect power of the index. Finally, extract the root of the factor that is a perfect power of the root.
$\bf{\text{Solution Details:}}$
Since $25=5^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{(5)^2(-3)^4(5)^3}
.\end{array}
Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{(5)^{2+3}(-3)^4}
\\\\=
\sqrt[3]{(5)^{5}(-3)^4}
.\end{array}
Factoring the expression that is a perfect power of the index and then extracting the root result to
\begin{array}{l}\require{cancel}
\sqrt[3]{(5)^{3}(-3)^3\cdot\left[(5)^2(-3)^1 \right]}
\\\\=
\sqrt[3]{\left[5\cdot(-3)\right]^3\cdot\left[25(-3) \right]}
\\\\=
5\cdot(-3)\sqrt[3]{25(-3)}
\\\\=
-15\sqrt[3]{-75}
.\end{array}
