# How do you rationalize the denominator and simplify 2/(5-sqrt3)?

May 18, 2018

See a solution process below:

#### Explanation:

To rationalize the denominator we must multiply the fraction by the appropriate form of $1$ to eliminate the radicals from the denominator.

For a denominator of the type $\left(a \textcolor{red}{-} b\right)$ we need to multiply by $\left(a \textcolor{red}{+} b\right)$:

$\frac{5 + \sqrt{3}}{5 + \sqrt{3}} \cdot \frac{2}{5 - \sqrt{3}} \implies$

$\frac{2 \left(5 + \sqrt{3}\right)}{\left(5 + \sqrt{3}\right) \cdot \left(5 - \sqrt{3}\right)} \implies$

$\frac{2 \cdot 5 + 2 \sqrt{3}}{\left(5 \cdot 5\right) - 5 \sqrt{3} + 5 \sqrt{3} - \sqrt{3} \sqrt{3}} \implies$

$\frac{10 + 2 \sqrt{3}}{25 - 0 - 3} \implies$

$\frac{10 + 2 \sqrt{3}}{22} \implies$

$\frac{5 + \sqrt{3}}{11} \implies$

Or

$\frac{5}{11} + \frac{\sqrt{3}}{11}$

May 18, 2018

$\text{ }$
color(brown)(2/(5-sqrt3) = [5+sqrt(3)]/11

#### Explanation:

$\text{ }$
Rationalize the denominator and simplify: color(blue)(2/(5-sqrt3)

"Given the rational expression: " color(red)(2/(5-sqrt3)

rArr 2/(5-sqrt3)*color(blue)([(5+sqrt3)/(5+sqrt3)]

$\Rightarrow \left[\frac{2 \cdot \left(5 + \sqrt{3}\right)}{\left({5}^{2}\right) - {\left(\sqrt{3}\right)}^{2}}\right]$

Identity used: :color(green)((a+b)(a-b)=a^2-b^2

In our problem: $a = 5 \mathmr{and} b = \sqrt{3}$

$\Rightarrow \frac{2 \cdot \left(5 + \sqrt{3}\right)}{25 - 3}$

$\Rightarrow \frac{2 \left(5 + \sqrt{3}\right)}{22}$

$\Rightarrow \frac{\cancel{2} \left(5 + \sqrt{3}\right)}{{\cancel{22}}^{\textcolor{b l u e}{11}}}$

$\Rightarrow \frac{5 + \sqrt{3}}{11}$

Hence,

color(brown)(2/(5-sqrt3) = [5+sqrt(3)]/11

Hope it helps.