Question

Find the value of a & b in the following
$5 + 2 \sqrt{3} \div 7 + 4 \sqrt{3} = a - b \sqrt{3}$
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Answer 1

$\sf given : - \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a - b \sqrt{3} \\ \\ \sf LHS : - \\ \\ \sf = \frac{5 + 2 \sqrt{ 3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ \sf = \frac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3} )}{(7 + 4 \sqrt{3} )(7 - 4 \sqrt{3} )} \\ \\ \sf = \frac{5(7 - 4 \sqrt{3} ) + 2 \sqrt{3}(7 - 4 \sqrt{3}) }{( {7)}^{2} - ( {4 \sqrt{3} })^{2} } \\ \\ \sf = \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24}{49 - 48} \\ \\ \sf = 11 - 6 \sqrt{3}$

comparing LHS with RHS we get,

➡ 11 - 6√3 = a - b√3

hence, a = 11 and b = 6

identity used :- (a + b)(a - b) = a² - b²

Answer 2

$\dfrac{5 \: + \: 2 \sqrt{3} }{7 \: + \: 4 \sqrt{3} } = \: a \: - \: b \sqrt{3}$

___________ [ GIVEN ]

• We have to find the value of a and b

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=> $\dfrac{5 \: + \: 2 \sqrt{3} }{7 \: + \: 4 \sqrt{3} } = \: a \: - \: b \sqrt{3}$

Rationalize it ..

=> $\dfrac{5 \: + \: 2 \sqrt{3} }{7 \: + \: 4 \sqrt{3} }$ × $\dfrac{7 \: - \: 4 \sqrt{3} }{7 \: - \: 4 \sqrt{3} }$

(a + b) (a - b) = a² - b²

=> $\dfrac{(5 \: + \: 2 \sqrt{3})(7 \: - \: 4 \sqrt{3}) }{ {(7)}^{2} \: - \:( 4 \sqrt{3})^{2} }$

=> $\dfrac{(5 \: + \: 2 \sqrt{3})(7 \: - \: 4 \sqrt{3}) }{ 49 \: - \:48 }$

=> $(5 \: + \: 2 \sqrt{3})(7 \: - \: 4 \sqrt{3})$

=> 5(7 - 4√3) + 2√3(7 - 4√3)

=> 35 - 20√3 + 14√3 - 24

=> 11 - 6√3

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Now ..

a - b√3 = 11 - 6√3

Here.. a = 11

and b√3 = 6√3

=> b = 6

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$\huge{a\:=\:11}$

and

$\huge{b\:=\:6}$

___________ [ ANSWER ]

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