Question

Simplify by shortcut not by using full division method......
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Answer 1

The answer is in attachment...

Explanation:

Each fraction is in the form of

$\displaystyle \frac{1}{ \sqrt{n} + \sqrt{n + 1} }$

Multiply each fraction by

$\sqrt{n + 1} - \sqrt{n}$

to rationalize the denominator according to the value of n there.

E.g.:

$\sqrt{2} - \sqrt{1} \ is \ mutiplied \ \: with \ \: \displaystyle \frac{1}{ \sqrt{1} + \sqrt{2} }$

After multiplying, each numerator will be in the form

$\sqrt{n + 1} - \sqrt{n}$

and the denominator will be in the form

$( \sqrt{n} + \sqrt{n + 1} )( \sqrt{n + 1} - \sqrt{n} )$

Multiply each factor in the denominator to get

$n + 1 - n = 1$

according to

$(a + b)(a - b) = a^{2} - {b}^{2}$

Thus there remains the one at the last step. Corresponding negative and positive values will be removed, and

$\sqrt{8} - \sqrt{1}$

remains there. Thus found!!!