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

1 Answer
Aviv S.
Mar 19, 2018

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!

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