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

1 Answer
Jun 12, 2016

Answer:

# = 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#