Question

If\[\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = x + y\sqrt{3},\]  find the values of x and y.

Answer 1

It is given that;

.  `(sqrt3-1)/ (sqrt3+1 )= x+ysqrt3`  we need to find x and y

We know that rationalization factor for  `sqrt3 +1` is`sqrt3 -1` . We will multiply numerator and denominator of the given expression `(sqrt3-1)/(sqrt3+1)`by,`sqrt3-1` to get

`(sqrt3-1)/ (sqrt3+1 ) xx (sqrt3-1)/(sqrt3-1) = ((sqrt3)^2 + (1) ^2 - 2 xx sqrt3 xx1) /((sqrt3)^2 - (1)^2)` 

`= (3+1-2sqrt3) /(3-1)`

` = (4-2sqrt3)/2`

` = 2-sqrt3`

On equating rational and irrational terms, we get 

  ` x + y sqrt3 = 2-sqrt3`

Hence, we get ` x= 2, y = -1`