1. ## Rationalizing a radical expression

Hello everyone I am needing some help with rationalizing and expression.

This is what I currently have:
$\frac{-5}{2(x-5)^2\sqrt{\frac{x}{(x-5}}}$

now I multiplied both sides by $\sqrt{\frac{x}{(x-5)}}$

got $\frac{-5 \sqrt{\frac{x}{x-5}}}{\frac{2x(x-5)^2}{(x-5)}}$

I canceled out the terms and got $\frac{-5 \sqrt \frac{x}{(x-5)}}{2x(x-5)}$

Now I have no idea what to do with the expression in the numerator. Any help would be appreciated and Thanks in advance!

2. ## Re: Rationalizing a radical expression

Obviously the denominator is done. Now think about what methods can be used to rationalize the x-5 term in the numerator, and select one that works.

3. ## Re: Rationalizing a radical expression

Hello, DjNito!

$\text{Simplify and rationalize: }\:\dfrac{-5}{2(x-5)^2\sqrt{\dfrac{x}{x-5}}}$

I would do it like this . . .

$\dfrac{-5}{2(x-5)^2\cdot\dfrac{x^{\frac{1}{2}}}{(x-5)^{\frac{1}{2}}}} \;=\;\dfrac{-5}{3(x-5)^{\frac{3}{2}}x^{\frac{1}{2}}}$

Multiply by $\frac{x^{\frac{1}{2}}(x-5)^{\frac{1}{2}}}{x^{\frac{1}{2}}(x-5)^{\frac{1}{2}}}:\;\; \dfrac{-5}{3(x-5)^{\frac{3}{2}}x^{\frac{1}{2}}} \cdot \frac{x^{\frac{1}{2}}(x-5)^{\frac{1}{2}}}{x^{\frac{1}{2}}(x-5)^{\frac{1}{2}}}$

. . . . . $=\; \dfrac{-5x^{\frac{1}{2}}(x-5)^{\frac{1}{2}}}{3x(x-5)^2} \;=\; \dfrac{-5\sqrt{x(x-5)}}{3x(x-5)^2}$

4. ## Re: Rationalizing a radical expression

Originally Posted by DjNito
Hello everyone I am needing some help with rationalizing and expression.

This is what I currently have:
$\frac{-5}{2(x-5)^2\sqrt{\frac{x}{(x-5}}}$

now I multiplied both sides by $\sqrt{\frac{x}{(x-5)}}$

got $\frac{-5 \sqrt{\frac{x}{x-5}}}{\frac{2x(x-5)^2}{(x-5)}}$

I canceled out the terms and got $\frac{-5 \sqrt \frac{x}{(x-5)}}{2x(x-5)}$

Now I have no idea what to do with the expression in the numerator. Any help would be appreciated and Thanks in advance!
\displaystyle \begin{align*} -\frac{5}{2 \left( x - 5 \right) ^2 \sqrt{ \frac{x}{x - 5} }} &= -\frac{5}{2(x-5)^2}\left( \frac{1}{\sqrt{\frac{x}{x-5}}} \right) \\ &= -\frac{5}{2(x - 5)^2} \left( \frac{1}{\frac{\sqrt{x}}{\sqrt{x-5}}} \right) \\ &= -\frac{5}{2(x-5)^2} \left( \frac{\sqrt{x-5}}{\sqrt{x}} \right) \\ &= -\frac{5\,\sqrt{x}\,\sqrt{x-5}}{2x \left(x-5 \right)^2} \end{align*}