How do you rationalize the denominator and simplify #8/(2sqrt x +3 )#?

1 Answer
Mar 19, 2018

Answer:

The fraction is equal to #(16sqrtx-24)/(4x-9)#.

Explanation:

The strategy is to multiply by the conjugate of the denominator. A conjugate of a two-term number looks like this:

The conjugate of #x+y# is #x-y#.

Multiplying the top and the bottom by the conjugate will cancel out the square roots of #x# on the bottom, leaving only #x#'s. It will look like this:

#color(white)=8/(2sqrtx+3)#

#=8/(2sqrtx+3)color(red)(*((2sqrtx-3))/((2sqrtx-3)))#

#=(8*(2sqrtx-3))/((2sqrtx+3)*(2sqrtx-3))#

#=(16sqrtx-24)/((2sqrtx+3)*(2sqrtx-3))#

#=(16sqrtx-24)/(2^2sqrtx^2-6sqrtx+6sqrtx-3*3)#

#=(16sqrtx-24)/(4xcolor(red)cancelcolor(black)(-6sqrtx+6sqrtx)-9)#

#=(16sqrtx-24)/(4x-9)#

The fraction is rationalized. Hope this helped!