Question

please solve it imidiately please I request

Answer 1

$\textbf{Solution:}$
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1) Represent 0.235 bar in the p/q form.

Let x = 0.235235......(1)

1000x = 235.235235.......(2)

On subtracting (1) from (2) we get;

1000x - x = 235 - 0

=> 999x = 235

=> x = $\frac{235}{999}$

Hence, it is in p/q form.

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2) Represent 0. 2113 bar on 3 in p/q form.

Let x = 0.2113333....(1)

100x = 211.33333....(2)

1000x = 2113.33333......(3)

On subtracting (2) from (3) we get;

1000x - 100x = 2113 - 211

=> 900x = 1902

=> x = $\frac{1902}{900}$

Hence, it is in the P / Q form.

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3)

$\frac{4}{ \sqrt{7} + \sqrt{3} } \\ \\ \frac{4}{ \sqrt{7} + \sqrt{3} } \times \frac{ \sqrt{7} - \sqrt{3} }{ \sqrt{7} - \sqrt{3} } \\ \\ \frac{4( \sqrt{7} - \sqrt{3}) }{( \sqrt{7} ) {}^{2} - ( \sqrt{3} ) {}^{2} } \\ \\ \frac{4 (\sqrt{7} - \sqrt{3} )}{7 - 3} \\ \\ \frac{4 (\sqrt{7} - \sqrt{3} )}{4} \\ \\ \sqrt{7} - \sqrt{3}$

Hence, √7 - √3 is the answer.

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4)
$\frac{4}{3 - \sqrt{5} } \\ \\ \frac{4}{3 - \sqrt{5} } \times \frac{3 + \sqrt{5} }{3 + \sqrt{5} } \\ \\ \frac{4(3 + \sqrt{5}) }{(3) {}^{2} - ( \sqrt{5} ) {}^{2} } \\ \\ \frac{4(3 + \sqrt{5}) }{4} = 3 + \sqrt{5}$

Therefore, the value of 0 + 4/a , is..

$\frac{4}{3 + \sqrt{5} } \\ \\ \frac{4}{3 + \sqrt{5} } \times \frac{3 - \sqrt{5} }{3 - \sqrt{5} } \\ \\ \frac{4(3 - \sqrt{5} )}{(3) {}^{2} - ( \sqrt{5} ) {}^{2} } \\ \\ \frac{4(3 - \sqrt{5}) }{4} = 3 - \sqrt{5}$


Hence, the value is ( 3 - √5 ).

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#thank you!! ☺☺