Question

(✓3+1/✓3-1)=a+b✓3 find value of a and b

Answer 1

$\dfrac{ \sqrt{3} + 1 }{ \sqrt{3} - 1 } = a + b \sqrt{3}$


$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: By \: \: Rationalizatio n \: ,$



$= > \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1} \times \frac{ \sqrt{3} + 1}{ \sqrt{3} + 1 } = a + b \sqrt{3} \\ \\ \\ \\ = > \frac{ (\sqrt{3} + 1) {}^{2} }{ (\sqrt{3} {)}^{2} - 1 {}^{2} } = a + b \sqrt{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{ | \: \: \: (a + b)(a - b) = {a}^{2} - {b}^{2} } \\ \\ \\ \\ = > \frac{3 + 1 + 2 \sqrt{3} }{3 - 1} = a + b \sqrt{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{ | \: \: \: ( {a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab }$



$= > \frac{4 + 2 \sqrt{3} }{2} = a + b \sqrt{3} \\ \\ \\ \\ = > \frac{2(2 + \sqrt{3}) }{2} = a + b \sqrt{3} \\ \\ \\ \\ = > 2 + \sqrt{3} = a + b \sqrt{3}$






Comparing values , we get

a = 2

b = 1