Question

Rationalise the denominator of √3+1/ √3-1

Answer 1

√3+1/√3-1
multiply and divide the denominator by √3+1 as in denominator (a-b)(a+b)=a^2-b^2 will be formed
therefore
√3+1/√3-1*√3+1/√3+1
=(√3+1)^2/(√3)^2-(1)^2
=(4+2√3)/2
=2+√3

Answer 2

Identities used :

${(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

Now,

$\frac{ \sqrt{3} + 1}{ \sqrt{3} - 1 }$

On rationalizing the denominator we get,

$= \frac{ \sqrt{3} + 1 }{ \sqrt{3} - 1} \times \frac{ \sqrt{3} + 1 }{ \sqrt{3} + 1 } \\ \\ = \frac{ {( \sqrt{3}) }^{2} + {(1)}^{2} + 2( \sqrt{3})(1) }{ {( \sqrt{3}) }^{2} - {(1)}^{2} } \\ \\ = \frac{3 + 1 + 2 \sqrt{3} }{3 - 1} \\ \\ = \frac{4 + 2 \sqrt{3} }{2} \\ \\ = \frac{2(2 + \sqrt{3}) }{2} \\ \\ = 2 + \sqrt{3}$