Question

How do you simplify `5/sqrt3 - 2/(3+sqrt3)` by rationalizing the denominator?

Answer 1

Here, the denominators of both the terms need to be rationalized. So let's rationalize them one by one.

First `color(red)(5/sqrt3`
To rationalize the denominator, we multiply the numerator as well as the denominator by `sqrt3`

`5/sqrt3*sqrt3/sqrt3`

= `(5*sqrt3)/(sqrt3*sqrt3)`

= `color(red)((5*sqrt3)/3`(As the denominator is a Rational Number, we have successfully simplified the first term)

The second term is `color(blue)(2/(3+sqrt3)`

To rationalize the denominator, we multiply the numerator as well as the denominator by its Conjugate : `3-sqrt3`

`(2/(3+sqrt3))*(3-sqrt3)/(3-sqrt3)`

= `(2*(3-sqrt3))/(3^2-(sqrt3)^2)`
We used the Identity `(a+b)(a-b)=a^2-b^2` in the Denominator

= `(2*(3-sqrt3))/(9-3)`

= `(2*(3-sqrt3))/6`

= `color(blue)((3-sqrt3)/3`

Now we can simplify the original expression `5/sqrt3 - 2/(3+sqrt3)`

= `color(red)((5*sqrt3)/3` `-` `color(blue)((3-sqrt3)/3`

=`(5*sqrt3 - 3 +sqrt3) / 3`

=`color(green)( ((4*sqrt3) - 3 ) / 3`