How to rationalize the denominator ?

Question

Hi, 

 

I have no idea how to solve and would appreciate if you can guide me step by step. So this what I need help rationalizing : 

 

(x—9) / (√x +3) 

Answer 1

Alex,

To rationalize such an expression only means to get the square root out of the denominator. To do so, we treat it like any other fraction, we multiply it by another fraction which is equivalent to one, but which has the desired effect on the denominator. If the entire denominator were under the radical, we would simply need to multiply by the same radical to cancel it out. But because part is and part is not, we need to use the conjugate of the denominator.

 

(x - 9)              (√x - 3)                x√x -3x - 9√x + 27     

—————  •  —————  =  ———————————

(√x + 3)           (√x - 3)                x + 3√x - 3√x - 9  

 

    x√x -3x - 9√x + 27

= ———————————

               x - 9 

 

I think you'll agree, that while it doesn't really look simpler, we have achieved the goal of getting the radical out of the denominator without changing the value.