Answer
$p^{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\dfrac{ p^{1/5}p^{7/10}p^{1/2}}{\left( p^{3} \right)^{-1/5}}
.$
$\bf{\text{Solution Details:}}$
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{ p^{1/5}p^{7/10}p^{1/2}}{p^{3\cdot\left(-\frac{1}{5} \right)}}
\\\\=
\dfrac{ p^{1/5}p^{7/10}p^{1/2}}{p^{-\frac{3}{5}}}
.\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}
\dfrac{ p^{\frac{1}{5}+\frac{7}{10}+\frac{1}{2}}}{p^{-\frac{3}{5}}}
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
p^{\frac{1}{5}+\frac{7}{10}+\frac{1}{2}-\left(-\frac{3}{5}\right)}
\\\\
p^{\frac{1}{5}+\frac{7}{10}+\frac{1}{2}+\frac{3}{5}}
\\\\
p^{\frac{2}{10}+\frac{7}{10}+\frac{5}{10}+\frac{6}{10}}
\\\\
p^{\frac{20}{10}}
\\\\
p^{2}
.\end{array}
