Question

find the value of A and B when root 7 + root 2 whole divided by root 7 minus root 2 is equal to a + b root 14 ​

Answer 1

ATQ:

$\rm \Longrightarrow \dfrac{\big{\sqrt{7}} + \big{\sqrt{2}}}{\big{\sqrt{7}} - \big{\sqrt{2}}} = a + b\sqrt{14}$

We are asked to find the value of 'a' & 'b'. For that, let's first rationalize the denominator. We can do so by multiplying the conjugate of (√7 - √2) which is (√7 + √2) to both the numerator and denominator.

$\rm \Longrightarrow \dfrac{\big{\sqrt{7}} + \big{\sqrt{2}}}{\big{\sqrt{7}} - \big{\sqrt{2}}} \ \times \dfrac{\big{\sqrt{7}} + \big{\sqrt{2}}}{\big{\sqrt{7}} + \big{\sqrt{2}}}$

Using the below identities we get:

(a + b)(a + b) = (a + b)²

(a + b)(a - b) = a² - b²

$\rm \Longrightarrow \dfrac{\Big( \sqrt{7} + \big{\sqrt{2}} \ \Big)^2}{\big(\sqrt{7}\big)^2 - \big(\sqrt{2}\big)^2}$

Using the identity (a + b)² = a² + b² + 2ab we get:

$\rm \Longrightarrow \dfrac{\big(\sqrt{7}\big)^2 + \big(\sqrt{2}\big)^2 + 2\big(\sqrt{7}\big)\big(\sqrt{2}\big)}{7 - 2}$

$\rm \Longrightarrow \dfrac{7 + 2 + 2\sqrt{14}}{5}$

$\rm \Longrightarrow \dfrac{9 + 2\sqrt{14}}{5}$

$\rm \Longrightarrow \dfrac{9}{5} + \dfrac{2}{5} \sqrt{14}$

$\rm \Longrightarrow a + b\sqrt{14}$

Therefore:

a = 9/5

b = 2/5