Problems Set at an Oral Examination.
Simplify the following expressions.
$\frac{1 - a^{- 1 / 2}}{1 + a^{1 / 2}} - \frac{a^{1 / 2} + 4^{- 1 / 2}}{a - 1} .$

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Solution

by taking lcm and solving we get

$\frac{1 - \frac{1}{\sqrt{a}}}{1 + \sqrt{a}} - \frac{\sqrt{a} + \frac{1}{\sqrt{4}}}{a - 1} = \frac{\sqrt{a} - 1}{\sqrt{a} + a} - \frac{2 \sqrt{a} + 1}{2 a - 2} = \frac{3 \sqrt{a} - 4 - 2 a}{2 \sqrt{a} (a - 1)}$

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x + 2 x^{2} + 3 x^{3} + . . . + n x^{n} .

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(1 - a^{2}) : ((\frac{1 - a^{3 / 2}}{1 - a^{1 / 2}} + a^{1 / 2}) (\frac{1 + a^{3 / 2}}{1 + a^{1 / 2}} - a^{1 / 2})) + 1.

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