Question

If 5+√3 /5-√3 =a+b, find rational number,a and b.

Answer 1

$\frac{5 + \sqrt{3} }{5 - \sqrt{3} } = a + b$




By Rationalization,



$\frac{5 + \sqrt{3} }{5 - \sqrt{3} } \times \frac{5 + \sqrt{3} }{5 + \sqrt{3} } \\ \\ \\ \\ \frac{ {(5 + \sqrt{3}) }^{2} }{ {(5)}^{2} - ( \sqrt{3} )^{2} } \\ \\ \\ \frac{25 + 3 + 10 \sqrt{3} }{25 - 3} \\ \\ \\ \frac{28 + 10 \sqrt{3} }{22} \\ \\ \\ \frac{14 + 5 \sqrt{2} }{11}$




On comparing values,


$\frac{14}{11} = a \\ \\ \\ \\ \\ \frac{5 \sqrt{3} }{11} = b$





I hope this will help you


(-:

Answer 2

Hello friends!!

Here is your answer :

$\frac{5 + \sqrt{3} }{5 - \sqrt{3} } = a + b$

First we have to rationalise the denominator..

$\frac{5 + \sqrt{3} }{5 - \sqrt{3} } \: \times \frac{5 + \sqrt{3} }{5 + \sqrt{3} }$

$\frac{(5 + \sqrt{3} )(5 + \sqrt{3}) }{(5 - \sqrt{3} )(5 + \sqrt{3} )}$

$\frac{{(5 + \sqrt{3})}^{2} }{(5 - \sqrt{3})(5 + \sqrt{3}) }$

Using identity

( x + y )² = x² + y² + 2xy

( x + y ) ( x - y ) = x² - y²

$\frac{ {(5)}^{2} + {( \sqrt{3}) }^{2} + 2(5)( \sqrt{3} )}{ {(5)}^{2} - {( \sqrt{3} )}^{2} }$

$\frac{25 + 3 + 10 \sqrt{3} }{25 - 3}$

$\frac{28 + 10 \sqrt{3} }{22}$

$\frac{28}{22} + \frac{10 \sqrt{3} }{22}$

$\frac{14}{11} + \frac{5 \sqrt{3} }{11}$

On comparing these values,

$a = \frac{14}{11}$

$b = \frac{5 \sqrt{3} }{11}$

Hope it helps you.. ☺️☺️☺️☺️

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