Answer
The simplified form of the expression $\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}$ is $\frac{5}{\sqrt{{{\left( 5-{{x}^{2}} \right)}^{3}}}}$.
Work Step by Step
Consider the provided expression, $\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}$
Multiply the numerator and denominator by $\sqrt{5-{{x}^{2}}}$.
Therefore,
$\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}=\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}\cdot \frac{\sqrt{5-{{x}^{2}}}}{\sqrt{5-{{x}^{2}}}}$
Apply the distributive property in the numerator: $a\left( b+c \right)=ab+ac$
Therefore,
$\begin{align}
& \frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}=\frac{\sqrt{5-{{x}^{2}}}\cdot \sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}\cdot \sqrt{5-{{x}^{2}}}}{\left( 5-{{x}^{2}} \right)\left( \sqrt{5-{{x}^{2}}} \right)} \\
& =\frac{{{\left( \sqrt{5-{{x}^{2}}} \right)}^{2}}+{{x}^{2}}}{\left( 5-{{x}^{2}} \right)\sqrt{5-{{x}^{2}}}}
\end{align}$
Apply the power of a power property ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ in the expression ${{\left( \sqrt{5-{{x}^{2}}} \right)}^{2}}$.
Therefore,
$\begin{align}
& {{\left( \sqrt{5-{{x}^{2}}} \right)}^{2}}={{\left( 5-{{x}^{2}} \right)}^{2\cdot \frac{1}{2}}} \\
& ={{\left( 5-{{x}^{2}} \right)}^{1}} \\
& =5-{{x}^{2}}
\end{align}$
Therefore,
$\begin{align}
& \frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}=\frac{5-{{x}^{2}}+{{x}^{2}}}{\left( 5-{{x}^{2}} \right)\sqrt{5-{{x}^{2}}}} \\
& =\frac{5}{\left( 5-{{x}^{2}} \right)\sqrt{5-{{x}^{2}}}}
\end{align}$
Further simplify,
$\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}=\frac{5}{{{\left( 5-{{x}^{2}} \right)}^{1}}{{\left( 5-{{x}^{2}} \right)}^{\frac{1}{2}}}}$
Apply,
The product property: ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$
Therefore,
$\begin{align}
& \frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}=\frac{5}{{{\left( 5-{{x}^{2}} \right)}^{1+\frac{1}{2}}}} \\
& =\frac{5}{{{\left( 5-{{x}^{2}} \right)}^{\frac{3}{2}}}} \\
& =\frac{5}{{{\left( {{\left( 5-{{x}^{2}} \right)}^{3}} \right)}^{\frac{1}{2}}}} \\
& =\frac{5}{\sqrt{{{\left( 5-{{x}^{2}} \right)}^{3}}}}
\end{align}$
Therefore, the simplified form of the expression $\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}$ is $\frac{5}{\sqrt{{{\left( 5-{{x}^{2}} \right)}^{3}}}}$.
