Question

How do you rationalize the denominator and simplify `3 / ( sqrt 5 + sqrt 6 )`?

Answer 1

`= 3sqrt6 -3sqrt5`

Explanation:

`3 / (sqrt5 + sqrt6`

We rationalise the denominator by multiplying the expression by the conjugate of the denominator. `color(blue)(sqrt5 - sqrt6`

`(3 * color(blue)((sqrt5 - sqrt6))) / ((sqrt5 + sqrt6) * color(blue)((sqrt5 - sqrt6)`

`= (3 * color(blue)((sqrt5)) + 3 * color(blue)( (- sqrt6))) / ((sqrt5 + sqrt6) * color(blue)((sqrt5 - sqrt6)`

`= (3sqrt5 - 3sqrt6) / ((sqrt5 + sqrt6) * color(blue)((sqrt5 - sqrt6)`

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

`= (3sqrt5 - 3sqrt6) / ((sqrt5)^2 - (sqrt6)^2)`

`= (3sqrt5 - 3sqrt6) / (5-6)`

`= (3sqrt5 - 3sqrt6) / (-1)`

`= -3sqrt5 + 3sqrt6`

`= 3sqrt6 -3sqrt5`