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

Question

Abigail Allen · Accepted Answer

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

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!

Abigail Allen · Answer

undefined To rationalize the denominator and simplify 8/(2√x + 3), we can multiply both the numerator and denominator by the conjugate of the denominator, which is 2√x - 3. This will eliminate the square root in the denominator.

By applying the conjugate, we get:
8 * (2√x - 3) / ((2√x + 3) * (2√x - 3))

Expanding the denominator:
8 * (2√x - 3) / (4x - 9)

Simplifying further, we have:
16√x - 24 / (4x - 9)