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#