Answer
$3,125x^{10}y^{35}z^{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
(5x^2y^7z)^{\frac{3}{2}}(5x^2y^7z)^{\frac{7}{2}}
,$ use the laws of exponents.
$\bf{\text{Solution Details:}}$
Using the Power of the Product 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}
(5x^2y^7z)^{\frac{3}{2}}(5x^2y^7z)^{\frac{7}{2}}
\\\\=
\left( 5^{\frac{3}{2}}x^{2\cdot\frac{3}{2}}y^{7\cdot\frac{3}{2}}z^{\frac{3}{2}} \right) \left( 5^{\frac{7}{2}}x^{2\cdot\frac{7}{2}}y^{7\cdot\frac{7}{2}}z^{\frac{7}{2}} \right)
\\\\=
\left( 5^{\frac{3}{2}}x^{3}y^{\frac{21}{2}}z^{\frac{3}{2}} \right) \left( 5^{\frac{7}{2}}x^{7}y^{\frac{49}{2}}z^{\frac{7}{2}} \right)
.\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}
\left( 5^{\frac{3}{2}}x^{3}y^{\frac{21}{2}}z^{\frac{3}{2}} \right) \left( 5^{\frac{7}{2}}x^{7}y^{\frac{49}{2}}z^{\frac{7}{2}} \right)
\\\\=
5^{\frac{3}{2}+\frac{7}{2}}x^{3+7}y^{\frac{21}{2}+\frac{49}{2}}z^{\frac{3}{2}+\frac{7}{2}}
\\\\=
5^{\frac{10}{2}}x^{10}y^{\frac{70}{2}}z^{\frac{10}{2}}
\\\\=
5^{5}x^{10}y^{35}z^{5}
\\\\=
3,125x^{10}y^{35}z^{5}
.\end{array}
