Question

find the valueof a and b √3-1/√3+1 =a+b√3​

Answer 1

Solution:-

Given

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

To find

$\rm \implies \: \: value \: \: a \: \: and \: \: b$

Now take

$\rm \implies \: \dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1 } \times \dfrac{ \sqrt{3} - 1}{ \sqrt{3} - 1}$

$\rm \implies \: \dfrac{( \sqrt{3} - 1) {}^{2} }{( \sqrt{3} + 1)( \sqrt{3} - 1)}$

Use this two identities

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

$\rm \implies \: (a - b)(a + b) = {a}^{2} - {b}^{2}$

Now we use

$\rm \implies \: \dfrac{( \sqrt{3} ) {}^{2} + {1}^{2} - 2 \sqrt{3} }{( \sqrt{3}) {}^{2} - {1}^{2} }$

$\rm \implies \: \dfrac{3 + 1 - 2 \sqrt{3} }{2}$

$\rm \implies \: \dfrac{4 - 2 \sqrt{3} }{2}$

$\rm \implies \: \dfrac{2(2 - \sqrt{3}) }{2} \implies \: 2 - \sqrt{3}$

Now compare with

$\rm\implies \: a + b \sqrt{3}$

We get the value of a = 2 and b = - 1