Question

Simplify 1/1+√2 + 1/√2+√3 + 1/√3+√4

Answer 1

★ To simplify :

$\dfrac{1}{1+ \sqrt{2}} + \dfrac{1}{\sqrt{2} + \sqrt{3}} + \dfrac{1}{\sqrt{3} + \sqrt{4}}$

★ Solution :

by rationalising

$\dfrac{1}{1 + \sqrt{2}} =$

$\dfrac{1}{1+\sqrt{2}} \times \dfrac{1 - \sqrt{2}}{1 - \sqrt{2}}$

$\dfrac{1 - \sqrt{2}}{(1 + \sqrt{2}) (1 - \sqrt{2}) }$

→ (a+b)(a-b) = a² - b²

$\implies \dfrac{1 - \sqrt{2}} {(1)^2 - (\sqrt{2})^2 }$

$\implies \dfrac{1 - \sqrt{2}}{1-2}$

$\implies \dfrac{1 - \sqrt{2}}{-1}$

→ -(1 - √2) = -1 + √2

in the same way rationalising other terms too

$\dfrac{1}{\sqrt{2}+ \sqrt{3}} = -\sqrt{2} + \sqrt{3}$

$\dfrac{1}{\sqrt{3} + \sqrt{4}} = -\sqrt{3} + \sqrt{4}$

by putting the rationalised values :

(-1 + √2) + (-√2 + √3) + (-√3 + √4)

= -1 + √2 - √2 + √3 - √3 + √4

= -1 + √4

4 = 2²

= -1 + √2²

= -1 + 2

= 1 (Answer)