How do you rationalize the denominator and simplify 5/(sqrt3-1)?

2 Answers
Aug 10, 2017

To rationalise this denominator, you have to take the denominator's conjugate. The conjugate will evaluate into a difference of two squares ( a^2 - b^2 = (a + b)(a - b) ) which can be expressed as an integer.

So to rationalise this expression, 5/(sqrt(3)-1)

5/(sqrt(3)-1)

= 5/(sqrt(3)-1) * (sqrt(3)+1)/(sqrt(3)+1)

= (5(sqrt(3)+1))/(3-1)

= (5sqrt(3) + 5)/(2)

Aug 10, 2017

color(green)((5(sqrt3+1))/2

Explanation:

:.5/(sqrt3-1) xx (sqrt3+1)/ (sqrt3+1)

sqrt3 xx sqrt3=3

color(white)(aaaaaaaaaaaaa)sqrt3-1
color(white)(aaaaaaaaaaa) xx underline(sqrt3+1)
color(white)(aaaaaaaaaaaaa)3-sqrt3
color(white)(aaaaaaaaaaaaaaaa)sqrt3-1
color(white)(aaaaaaaaaaaaa)overline(3+0-1)

color(white)(aaaaaaaaaaaaa)color(green)(=2

:.color(green)((5(sqrt3+1))/2

~~~~~~~~~~~~~~~~~~~~~

check by calculator:

:.5/(sqrt3-1)=color(green)(6.830127019

:.(5(sqrt3+1))/2=color(green)(6.830127019