Answer
The simplified value of the expression is $\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{{{\left( 5-x \right)}^{2}}{{\left( 5+x \right)}^{2}}}$.
Work Step by Step
Consider the expression $\frac{\sqrt{25-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{25-{{x}^{2}}}}}{25-{{x}^{2}}}$.
Here, the least common denominator is $\sqrt{25-{{x}^{2}}}$.
So, multiply the numerator and denominator by $\sqrt{25-{{x}^{2}}}$ ,
That is,
$\begin{align}
& \frac{\sqrt{25-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{25-{{x}^{2}}}}}{25-{{x}^{2}}}\cdot \frac{\sqrt{25-{{x}^{2}}}}{\sqrt{25-{{x}^{2}}}}=\frac{\sqrt{25-{{x}^{2}}}\cdot \sqrt{25-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{25-{{x}^{2}}}}\cdot \sqrt{25-{{x}^{2}}}}{25-{{x}^{2}}\cdot \sqrt{25-{{x}^{2}}}} \\
& =\frac{25-{{x}^{2}}+{{x}^{2}}}{{{\left( 25-{{x}^{2}} \right)}^{1}}\cdot {{\left( 25-{{x}^{2}} \right)}^{\frac{1}{2}}}} \\
& =\frac{25}{{{\left( 25-{{x}^{2}} \right)}^{1+\frac{1}{2}}}} \\
& =\frac{25}{{{\left( 25-{{x}^{2}} \right)}^{\frac{3}{2}}}}
\end{align}$
Further simplify the above expression:
$\begin{align}
& \frac{\sqrt{25-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{25-{{x}^{2}}}}}{25-{{x}^{2}}}\cdot \frac{\sqrt{25-{{x}^{2}}}}{\sqrt{25-{{x}^{2}}}}=\frac{25}{\sqrt{{{\left( 25-{{x}^{2}} \right)}^{3}}}} \\
& =\frac{25}{\sqrt{{{\left( 25-{{x}^{2}} \right)}^{3}}}}\cdot \frac{\sqrt{\left( 25-{{x}^{2}} \right)}}{\sqrt{\left( 25-{{x}^{2}} \right)}} \\
& =\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{\left( 25-{{x}^{2}} \right)\left( 25-{{x}^{2}} \right)} \\
& =\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{\left( {{5}^{2}}-{{x}^{2}} \right)\left( {{5}^{2}}-{{x}^{2}} \right)}
\end{align}$
Apply the difference of the square property in the denominator as shown:
$\begin{align}
& \frac{\sqrt{25-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{25-{{x}^{2}}}}}{25-{{x}^{2}}}\cdot \frac{\sqrt{25-{{x}^{2}}}}{\sqrt{25-{{x}^{2}}}}=\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{\left( {{5}^{2}}-{{x}^{2}} \right)\left( {{5}^{2}}-{{x}^{2}} \right)} \\
& =\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{\left( 5-x \right)\left( 5+x \right)\left( 5-x \right)\left( 5+x \right)} \\
& =\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{{{\left( 5-x \right)}^{2}}{{\left( 5+x \right)}^{2}}}
\end{align}$
Hence, the simplified value of the expression is $\frac{25\sqrt{\left( 25-{{x}^{2}} \right)}}{{{\left( 5-x \right)}^{2}}{{\left( 5+x \right)}^{2}}}$.
