Question

if 2+√3/ 2-√3 = a-b√3 then find a and b​

Answer 1

Given

$\sf \to \: \dfrac{2 + \sqrt{3} }{2 - \sqrt{3} } = a - b \sqrt{3}$

To Find the value of a and b

So ,Take

$\sf \to \: \dfrac{2 + \sqrt{3} }{2 - \sqrt{3} }$

Rationalize The Denominator

$\sf \to \: \dfrac{2 + \sqrt{3} }{2 - \sqrt{3} } \times \dfrac{2 + \sqrt{3} }{2 + \sqrt{3} }$

We Know that

$\sf \to \: (a + b)(a + b) = (a + b) {}^{2}$

$\sf \to \: (a - b)(a + b) = {a}^{2} - {b}^{2}$

We get

$\sf \to \: \dfrac{(2 + \sqrt{3}) {}^{2} }{( {2}^{2} - ( \sqrt{3}) {}^{2} )}$

Using this identities

$\sf \to \: (a + b) {}^{2} = {a}^{2} + {b}^{2} + 2ab$

we get

$\sf \to \: \dfrac{ {2}^{2} + ( \sqrt{3} ) {}^{2} - 2 \times 2 \times \sqrt{3} }{4 - 3}$

$\sf \to \: \dfrac{4 + 3 - 4 \sqrt{3} }{1}$

$\sf \to \: 7 - 4 \sqrt{3}$

By comparing with

$\sf \to \: a - b \sqrt{3}$

we get

$\sf \to \: a = 7 \: \: and \: b = 4$

Answer

$\sf \to \: a = 7 \: \: and \: b = 4$