Question

How do you rationalize the denominator `2/(5-sqrt3)`?

Answer 1

`2/(5-sqrt3)`

We can ratioinalise the denominator by multiplying the numerator and denominator of the expression by the conjugate of `5-sqrt3`

The conjugate of `5-sqrt3 = color(red)(5+sqrt3`

`=(2xxcolor(red)((5+sqrt3)))/((5-sqrt3)xxcolor(red)((5+sqrt3))`

We know that `color(blue)((a-b)(a+b)=a^2 - b^2`
so ,`(5-sqrt3)xx(5+sqrt3) = 5^2- sqrt3^2 = color(blue)(25 - 3 = 22`

the expression becomes:
`(cancel2xxcolor(red)((5+sqrt3)))/cancel22`

`= (5+ sqrt3)/ 11`