Question

rationalize the demominater and find the value of a&b
$\frac{2 + \sqrt{3} }{2 - \sqrt{3} } = a + b \sqrt{3}$
​

Answer 1

We have,

• To find the value of a & b in the (above) expression

Given,

$\boxed{ \boxed{\Large{\rm \frac{2 + \sqrt{3} }{2 - \sqrt{3} } = a + b \sqrt{3} }}}$

Now,

• Solving LHS,

$\boxed{\dag }\: \: \: \: \: \frac{2 + \sqrt{3} }{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } \\ \\ => \frac{ {(2 + \sqrt{3} )}^{2} }{ {(2)}^{2} - {( \sqrt{3} )}^{2} } \: \: = \frac{4 + 3 + 4 \sqrt{3} }{4 - 3} \\ \\ = > \frac{7 + 4 \sqrt{3} }{1} \: \: \: = \underline{ \: 7 + 4 \sqrt{3} \: }$

Now,

• Comparing LHS to RHS, we conclude that,

$\bullet \: \boxed{ \bf \text{Value of \:a} = \: \: \: \red4 \: }\\ \bullet \: \boxed{ \bf \text{Value of \:b} = \red{- 3}}$

Thus,

$\rm \: \: \: \: \: \: \: \: \: \frac{2 + \sqrt{3} }{2 - \sqrt{3} } = 4+ ( - 3)\sqrt{3}$