Question

3 + √2 / 3 - √2 = a+b √2 . Find a and b

Answer 1

Answer:

$\bigstar{\bold{a=\dfrac{11}{7} }}$

$\bigstar{\bold{b=\dfrac{6}{7} }}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

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

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• a and b

$\Large{\underline{\underline{\bf{Solution:}}}}$

→ Rationalise the denominator by taking the conjugate of 3 - √2 which is equal to 3 + √2

→ Multiplying by 3 + √2 on both numerator and denominator

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

→ By using the identities

(a + b)² = a² + 2ab + b²

(a + b) (a -b) = a² - b²

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

$\dfrac{9+2+6\sqrt{2} }{9-2}$

$\dfrac{11+6\sqrt{2} }{7}$

→ Splitting it we get,

$\dfrac{11}{7} +\dfrac{6\sqrt{2} }{7}$

→ Equating it with a + b√2

a = 11/7

b = 6/7

$\boxed{\bold{a=\dfrac{11}{7}}}$     ,     $\boxed{\bold{b=\dfrac{6}{7}}}$

$\Large{\underline{\underline{\bf{Notes:}}}}$

→ (a + b)² = a² + 2ab + b²

→ (a - b)² = a² - 2ab + b²

→ a² - b² = (a +b) × (a - b)