Question

Find The Answer Of This Question ​

Answer 1

$\large\underline{\sf{Given- }}$

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

$\large\underline{\sf{To\:Find - }}$

The value of a and b.

$\large\underline{\sf{Solution-}}$

Given expression is

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

On rationalizing the denominator, we get

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

We know,

$\boxed{ \bf{ \: (x + y)(x - y) = {x}^{2} - {y}^{2}}}$

So, using this identity, we get

$\rm :\longmapsto\:\dfrac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24}{ {(7)}^{2} - {(4 \sqrt{3} )}^{2} } = a + b \sqrt{3}$

$\rm :\longmapsto\:\dfrac{11 - 6 \sqrt{3} }{ 49 - 48 } = a + b \sqrt{3}$

$\rm :\longmapsto\:\dfrac{11 - 6 \sqrt{3} }{ 1 } = a + b \sqrt{3}$

$\rm :\longmapsto\: \pink{11} - \purple{6 \sqrt{3} }= \pink{ a }+ \purple{b \sqrt{3} }$

So, On comparing we get

$\: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \bf{ \: a = 11}} \: \: \: and \: \: \: \boxed{ \bf{ \: b = - 6}}$

Additional Information :-

More Identities to know

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)