Question

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

Answer 1

`= (5 - sqrt7) / 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)`

`= (5 - sqrt7) / (25 - 7)`

`= (5 - sqrt7) / 18`