# Solve. x^\frac{-1}{2}(x + 9)^\frac{1}{2} + x^\frac{1}{2}(x + 9)^\frac{-1}{2}

## Question:

Solve.

{eq}x^\frac{-1}{2}(x + 9)^\frac{1}{2} + x^\frac{1}{2}(x + 9)^\frac{-1}{2} {/eq}

## Addition of nth Root Functions:

For the addition of the nth root function, first, we'll simplify the negative exponents by rewriting them as positive exponents functions.

{eq}\displaystyle x^{-\frac{1}{n}}=\frac{1}{x^{\frac{1}{n}}} {/eq}

After that, we'll use the most general product rule of exponents (when the base is the same ) shown below:

{eq}\displaystyle x^{m}\cdot x^{n}=x^{m+n} {/eq}

Given:

The given expression with different nth root functions is:

{eq}\displaystyle x^\frac{-1}{2}(x + 9)^\frac{1}{2} + x^\frac{1}{2}(x + 9)^\frac{-1}{2} {/eq}

Simplifying the above expression using negative exponents rule, we get:

{eq}\begin{align*} \displaystyle x^\frac{-1}{2}(x + 9)^\frac{1}{2} + x^\frac{1}{2}(x + 9)^\frac{-1}{2}&=\displaystyle \frac{1}{x^\frac{1}{2}}\cdot (x + 9)^\frac{1}{2} + x^\frac{1}{2}\cdot \frac{1}{(x + 9)^\frac{1}{2}}\\ &=\displaystyle \frac{(x + 9)^\frac{1}{2}}{x^\frac{1}{2}} +\frac{x^\frac{1}{2}}{(x + 9)^\frac{1}{2}}\\ \end{align*} {/eq}

Further, simplify the above expression using the product rule of exponents.

{eq}\begin{align*} \displaystyle \frac{(x + 9)^\frac{1}{2}}{x^\frac{1}{2}} +\frac{x^\frac{1}{2}}{(x + 9)^\frac{1}{2}} &=\displaystyle\frac{(x + 9)^\frac{1}{2}\cdot (x + 9)^\frac{1}{2}+x^\frac{1}{2}\cdot x^\frac{1}{2}}{x^\frac{1}{2}(x + 9)^\frac{1}{2}}\\ &=\displaystyle\frac{(x + 9)^{\frac{1}{2}+\frac{1}{2}}+x^{\frac{1}{2}+\frac{1}{2}}}{\sqrt{x}\sqrt{(x + 9)}}\\ &=\displaystyle\frac{(x + 9)^{\frac{2}{2}}+x^{\frac{2}{2}}}{\sqrt{x(x + 9)}}&\because \sqrt{a}\cdot \sqrt{b}=\sqrt{ab}\\ &=\displaystyle\frac{(x + 9)^{1}+x^{1}}{\sqrt{x^2 + 9x}}\\ &=\displaystyle\frac{x + 9+x}{\sqrt{x^2 + 9x}}\\ &=\displaystyle\boxed{\frac{2x + 9}{\sqrt{x^2 + 9x}}}\\ \end{align*} {/eq}