JAMB - Mathematics (1980 - No. 5)
Rationalize the denominator of the given expression \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\)
1 + 2a - 2\(\sqrt{a(1 + a)}\)
\(\sqrt{1(1 + a)}\)
2a - 2\(\sqrt{a(1 + a)}\)
1 + 2a - 2\(\sqrt{a + b}\)
Explanation
\(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\) = \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\) x \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} - \sqrt{a}}\)
= \(\frac{\sqrt{1 + a + a}}{1 + a - a}\)
= 2a + a(1 + a)
= 1 + 2a - 2\(\sqrt{a(1 + a)}\)
= \(\frac{\sqrt{1 + a + a}}{1 + a - a}\)
= 2a + a(1 + a)
= 1 + 2a - 2\(\sqrt{a(1 + a)}\)
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