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

1 Answer
Jun 12, 2016

= 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