Answer
The required expression is $\underline{\frac{-10}{{{\left( x+5 \right)}^{\frac{1}{2}}}{{\left( x-5 \right)}^{\frac{3}{2}}}}}$.
Work Step by Step
Consider the given polynomial:
${{\left( x-5 \right)}^{\frac{-1}{2}}}{{\left( x+5 \right)}^{\frac{-1}{2}}}-{{\left( x+5 \right)}^{\frac{1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}}$ ………..….. (1)
Apply the technique, ‘‘factoring the polynomial by greatest common factor.’’
$\begin{align}
& {{\left( x-5 \right)}^{\frac{-1}{2}}}{{\left( x+5 \right)}^{\frac{-1}{2}}}-{{\left( x+5 \right)}^{\frac{1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}}={{\left( x+5 \right)}^{\frac{-1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}}\left( {{\left( x-5 \right)}^{\frac{-1}{2}-\left( \frac{-3}{2} \right)}}-{{\left( x+5 \right)}^{\frac{1}{2}-\left( \frac{-1}{2} \right)}} \right) \\
& ={{\left( x+5 \right)}^{\frac{-1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}}\left( \left( x-5 \right)-\left( x+5 \right) \right) \\
& ={{\left( x+5 \right)}^{\frac{-1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}}\left( x-5-x-5 \right)
\end{align}$
Further solve,
$\begin{align}
& {{\left( x-5 \right)}^{\frac{-1}{2}}}{{\left( x+5 \right)}^{\frac{-1}{2}}}-{{\left( x+5 \right)}^{\frac{1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}}=-10{{\left( x+5 \right)}^{\frac{-1}{2}}}{{\left( x-5 \right)}^{\frac{-3}{2}}} \\
& =\frac{-10}{{{\left( x+5 \right)}^{\frac{1}{2}}}{{\left( x-5 \right)}^{\frac{3}{2}}}}
\end{align}$
