How do you divide #(4 + sqrt2)/(3sqrt3 - sqrt6)#?

1 Answer
Mar 24, 2015

Simplify the denominator by remembering that
#(a - b) xx (a + b) = (a^2 - b^2)#
and therefore we can get rid of the roots in the denominator by multiplying by #3sqrt(3) + sqrt(6)#

Of course if we multiply the denominator by #3sqrt(3) + sqrt(6)# we will have to multiply the numerator by that amount as well.

#(4 + sqrt(2))/(3sqrt(3)-sqrt(6)) * (3sqrt(3)+sqrt(6))/(3sqrt(3)+sqrt(6))#

#= (4(3sqrt(3)+sqrt(6)) + sqrt(2)(3sqrt(3)+sqrt(6)))/(27 - 6)#

#= (12 sqrt(3) + 4 sqrt(6) + 3sqrt(6) + sqrt(12))/21#

#= (12sqrt(3) + 7sqrt(6) + sqrt(12))/21#

or
#sqrt(3) (12 + 7sqrt(2) + sqrt(4))/21#

#= sqrt(3) (14 + 7sqrt(2))/21#

#= 7sqrt(3) * (2 + sqrt(2))/3#

or, possibly
#= 7 (2+sqrt(2))/sqrt(3)#