Question

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

Answer 1

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!