Precalculus (6th Edition) Blitzer

The simplified form of the expression $\frac{\sqrt{5-{{x}^{2}}}+\frac{{{x}^{2}}}{\sqrt{5-{{x}^{2}}}}}{5-{{x}^{2}}}$ is $\frac{5}{\sqrt{{{\left( 5-{{x}^{2}} \right)}^{3}}}}$.
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 $\frac{5}{\sqrt{{{\left( 5-{{x}^{2}} \right)}^{3}}}}$.