How do you rationalize the denominator & simplify $(5 + sqrt3)/(2 -sqrt3)$ ?

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Answer 1 · 2015-04-16T10:35:55

To rationalise the denominator, we multiply the Numerator and the Denominator of this fraction with the Conjugate of the Denominator

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

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

The Denominator is in the form $color(blue)((a-b)*(a+b)$ which equals $color(blue)(a^2 - b^2$

$= ((5 + sqrt3)(2+sqrt3)) /( 2^2 - (sqrt 3)^2)$

$= ((5 + sqrt3)(2+sqrt3)) /( 4 - 3)$

$= ((5 + sqrt3)(2+sqrt3)) / 1$

$= (5 + sqrt3)(2+sqrt3)$

Using the Distributive Property of Multiplication we get:

$= (5*2) + 5sqrt3 + 2sqrt 3 + (sqrt 3 * sqrt 3)$

$= 10 + 7sqrt 3 + 3$

$color(green)(= 13 + 7 sqrt 3$

How do you rationalize the denominator & simplify (5 + sqrt3)/(2 -sqrt3)(5 + sqrt3)/(2 -sqrt3)?