Question

answer fast plz plea​

Answer 1

Given equation:

$\sf\dfrac{3 + \sqrt{7} }{3 - \sqrt{7} } = a + b \sqrt{7}$

To find:

• Value of a and b

Solution:

By rationalising the LHS of given equation, (i.e. by multiplying both numerator and denominator of LHS with the rationalising factor which is 3 + √7), we get:

$\sf \implies\dfrac{3 + \sqrt{7} }{3 - \sqrt{7} } \times \dfrac{3 + \sqrt{7} }{3 + \sqrt{7} } = a + b \sqrt{7}$

$\sf \implies\dfrac{(3 + \sqrt{7} )^2}{(3 - \sqrt{7} )(3 + \sqrt{7}) } = a + b \sqrt{7}$

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Now, use the following identities:

• $\boxed{\tt (A+B)^2 = A^2 + B^2 + 2AB}$
• $\boxed{\tt (A+B)(A-B) = A^2 - B^2}$

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$\sf \implies\dfrac{(3)^{2} + (\sqrt{7} )^2 + 2(3)( \sqrt{7}) }{{(3)}^{2} - {( \sqrt{7} )}^{2}} = a + b \sqrt{7}$

$\sf \implies\dfrac{9 + 7 + 6\sqrt{7} }{9 - 7} = a + b \sqrt{7}$

$\sf \implies\dfrac{16 + 6\sqrt{7} }{2} = a + b \sqrt{7}$

$\sf \implies\dfrac{ \not2(8 + 3\sqrt{7}) } {\not2} = a + b \sqrt{7}$

$\sf \implies8 + 3\sqrt{7} = a + b \sqrt{7}$

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On comparing LHS and RHS, we get:

• a = 8
• b = 3

Therefore, option (c) is correct.

Answer 2

Answer :-

• Hence, option (c) is correct.

Step by step explanation :-

Given expression :

$\tt{\frac{3 + \sqrt{7} }{3 - \sqrt{7} } = a + b \sqrt{7} }$

First we rationalize the denominator,

$\rm:\longmapsto{ \frac{3 + \sqrt{7} }{3 - \sqrt{7} } \times \frac{3 + \sqrt{7} }{3 + \sqrt{7} } } \\ \\\rm:\longmapsto{ \frac{(3 + \sqrt{7}) {}^{2} }{(3 - \sqrt{7})(3 + \sqrt{7} ) } }$

Using (a+b)² identity in numerator and (a+b) (a-b) identity use in denominator,

$\rm:\longmapsto{ \frac{(3) {}^{2} + 2(3)( \sqrt{7} ) + ( \sqrt{7}) {}^{2} }{(3) {}^{2} - ( \sqrt{7} ) {}^{2} } } \\ \\ \rm:\longmapsto{ \frac{9 + 6 \sqrt{7} + 7}{9 - 7} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\\rm:\longmapsto{ \frac{16 + 6 \sqrt{7} }{2} = \frac{ \cancel2(8 + 3 \sqrt{7})}{ \cancel2} } \\ \\ \rm:\longmapsto{8 + 3 \sqrt{7} } = a + b \sqrt{7} \: \: \: \: \: \: \: \: \:$

Comparing LHS and RHS we get,

• a = 8
• b = 3

Hence, option (c) is correct.

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