# How do you rationalize the denominator and simplify 1/(5+sqrt7)?

Apr 5, 2016

$= \frac{5 - \sqrt{7}}{18}$

#### Explanation:

1 / ( 5 +sqrt7

We rationalise the expression by multiplying it by the conjugate of the denominator.  color(blue)( ( 5 - sqrt7)

 = (1 * color(blue)(( 5 - sqrt7))) / (( 5 +sqrt7) * color(blue)( ( 5 - sqrt7))

 = (1 * color(blue)(( 5)) + 1 * color(blue)(( - sqrt7))) / (( 5 +sqrt7) * color(blue)( ( 5 - sqrt7))

 = (5 - sqrt7) / (( 5 +sqrt7) * color(blue)( ( 5 - sqrt7))

• Applying property:
color(blue)((a+b)(a-b) = a^2 - b^2 to the denominator.

 = (5 - sqrt7) / (( 5 ^2 - (sqrt7)^2)

$= \frac{5 - \sqrt{7}}{25 - 7}$

$= \frac{5 - \sqrt{7}}{18}$