# How do you rationalize the denominator and simplify 4/(5+sqrt2)?

Mar 19, 2016

$= \frac{20 - 4 \sqrt{2}}{23}$

#### Explanation:

Consider ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Multiply by 1 but where $1 = \frac{5 - \sqrt{2}}{5 - \sqrt{2}}$

color(brown)(4/(5+sqrt(2))xx(5-sqrt(2))/(5-sqrt(2)))color(blue)(->(20-4sqrt(2))/(5^2-2)

$= \frac{20 - 4 \sqrt{2}}{23}$

Mar 19, 2016

The simplified solution is $\frac{4 \left(5 - \sqrt{2}\right)}{23}$

#### Explanation:

To rationalize the denominator you need to multiply it by a term that will eliminate the surd
You must then multiply the numerator by the same term to keep the value of the fraction the same.
So multiply by $\frac{5 - \sqrt{2}}{5 - \sqrt{2}}$
this gives

(4(5 - sqrt2))/( (5 + sqrt2)(5 - sqrt2) = $\frac{4 \left(5 - \sqrt{2}\right)}{25 - 2}$

So the solution is $\frac{4 \left(5 - \sqrt{2}\right)}{23}$