Question

solve these questions correct ly​

Answer 1

I m not familiar of rationalize. . .

hope remaining two answer is helpful for you. . .

Answer 2

(1).
$lhs = \frac{{ a}^{ - 1} }{{ a}^{ - 1} + { b}^{ - 1} } + \frac{{ a}^{ - 1} }{{ a}^{ - 1} - \: { b}^{ - 1} }$

$lhs = \frac{ \frac{1}{a} }{\frac{1}{a} + \frac{1}{b} } + \frac{ \frac{1}{a} }{\frac{1}{a} - \frac{1}{b} }$

$lhs = \frac{b}{ b + a} + \frac{b}{b - a}$

$lhs = \frac{ {b}^{2} - ab {b}^{2} + ab}{ {b}^{2} - {a}^{2} }$

$lhs = \frac{ 2{b}^{2} }{ {b}^{2} - {a}^{2} } = rhs$

(2).
$let \: {x}^{a - b} = y$
So,
$lhs = \frac{1}{1 + y} + \frac{1}{1 + \frac{1}{y} }$

$lhs = \frac{1}{1 + y} + \frac{y}{y + 1 }$

$lhs = \frac{1 + y}{1 + y} = 1 = rhs$

(3).

First multiply √3 - √2 - 5 in numerator and denominator.

Then multiply √6 - 4 in numerator and denominator.