Question

find the value of a and b
please help me to solve it
don't type irtrravalent​

Answer 1

Answer:

${\bf (i)} \: a = 5 \: {\rm and }\: b = 2$

${\bf (ii)} \: a = -\dfrac{19}{7}\:{\rm and }\: b = \dfrac{5}{7}$

Explanation:

Solving (i) :-

$\implies \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } = a + b \sqrt{6}$

Rationalise the denominator by taking L. H. S.

$= \dfrac{( \sqrt{3} + \sqrt{2} )}{( \sqrt{3} - \sqrt{2} )}$

$= \dfrac{( \sqrt{3} + \sqrt{2} )}{( \sqrt{3} - \sqrt{2} )} \times \dfrac{( \sqrt{3} + \sqrt{2} )}{( \sqrt{3} + \sqrt{2} )}$

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

$= \dfrac{3 + \sqrt{6} + \sqrt{6} + 2 }{ {( \sqrt{3} )}^{2} - {( \sqrt{2} )}^{2} }$

$= \dfrac{5 + 2 \sqrt{6} }{1}$

$= 5 + 2 \sqrt{6}$

On comapring with R. H. S. :-

$\implies \: 5 + 2 \sqrt{6} = a + b \sqrt{6}$

$\therefore \: a = 5 \: { \rm \: and} \: b = 2$

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Solving (ii) :-

$\implies \: \dfrac{( \sqrt{5} + \sqrt{3}) }{(2 \sqrt{5} - 3 \sqrt{3} )} = a - b \sqrt{15}$

Rationalise the denominator of L. H. S. :-

$= \dfrac{( \sqrt{5} + \sqrt{3} )}{(2 \sqrt{5} - 3 \sqrt{3} )} \times \dfrac{(2 \sqrt{5} + 3 \sqrt{3} )}{(2 \sqrt{5} + 3 \sqrt{3})}$

By the identity :- (a - b)(a + b) = a² - b²

$= \dfrac{19 + 5 \sqrt{15} }{ {(2 \sqrt{5} )}^{2} - {(3 \sqrt{3}) }^{2} }$

$= \dfrac{19 + 5 \sqrt{15} }{20 - 27}$

$= \dfrac{19 + 5 \sqrt{15} }{-7}$

On comparing with R. H. S. :-

$\implies \dfrac{19 + 5\sqrt{15}}{-7} = a - b \sqrt{15}$

$\implies \dfrac{-19}{7} - \dfrac{5}{7}\sqrt{15} = a - b\sqrt{15}$

$\therefore \: a = -\dfrac{19}{7}\:{\rm and }\: b = \dfrac{5}{7}$