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

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Answer 1 · 2015-04-09T13:28:36

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 $5/sqrt3 - 2/(3+sqrt3)$

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

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

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

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