Question

Find the value of a and b
5+√3 / 7+2√3 = a-b√3

Answer 1

Answer:

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Answer 2

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

Given that

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

On rationalizing the denominator, we get

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

$\rm :\longmapsto\:\dfrac{35 - 10 \sqrt{3} + 7\sqrt{3} - 6}{ {7}^{2} - {(2 \sqrt{3} )}^{2} } = a - b \sqrt{3}$

$\red{\bigg \{ \because \: (x + y)( x - y) = {x}^{2} - {y}^{2} \bigg \}}$

$\rm :\longmapsto\:\dfrac{29 - 3 \sqrt{3}}{49- 12 } = a - b \sqrt{3}$

$\rm :\longmapsto\:\dfrac{29 - 3 \sqrt{3}}{37} = a - b \sqrt{3}$

$\rm :\longmapsto\: \dfrac{29}{37} - \dfrac{3 \sqrt{3}}{37} = a - b \sqrt{3}$

$\rm :\longmapsto\: \dfrac{29}{37} - \dfrac{3}{37} \sqrt{3} = a - b \sqrt{3}$

On comparing, we get

$\bf :\longmapsto\:a = \dfrac{29}{37} \: \: and \: \: b = \dfrac{3}{37}$

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)