How do you rationalize the denominator and simplify 8/(2sqrt x +3 )?

Mar 19, 2018

The fraction is equal to $\frac{16 \sqrt{x} - 24}{4 x - 9}$.

Explanation:

The strategy is to multiply by the conjugate of the denominator. A conjugate of a two-term number looks like this:

The conjugate of $x + y$ is $x - y$.

Multiplying the top and the bottom by the conjugate will cancel out the square roots of $x$ on the bottom, leaving only $x$'s. It will look like this:

$\textcolor{w h i t e}{=} \frac{8}{2 \sqrt{x} + 3}$

$= \frac{8}{2 \sqrt{x} + 3} \textcolor{red}{\cdot \frac{\left(2 \sqrt{x} - 3\right)}{\left(2 \sqrt{x} - 3\right)}}$

$= \frac{8 \cdot \left(2 \sqrt{x} - 3\right)}{\left(2 \sqrt{x} + 3\right) \cdot \left(2 \sqrt{x} - 3\right)}$

$= \frac{16 \sqrt{x} - 24}{\left(2 \sqrt{x} + 3\right) \cdot \left(2 \sqrt{x} - 3\right)}$

$= \frac{16 \sqrt{x} - 24}{{2}^{2} {\sqrt{x}}^{2} - 6 \sqrt{x} + 6 \sqrt{x} - 3 \cdot 3}$

$= \frac{16 \sqrt{x} - 24}{4 x \textcolor{red}{\cancel{\textcolor{b l a c k}{- 6 \sqrt{x} + 6 \sqrt{x}}}} - 9}$

$= \frac{16 \sqrt{x} - 24}{4 x - 9}$

The fraction is rationalized. Hope this helped!