Rationalize the denominator of the given expression \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\)
A
1 + 2a - 2\(\sqrt{a(1 + a)}\)
B
\(\sqrt{1(1 + a)}\)
C
2a - 2\(\sqrt{a(1 + a)}\)
D
1 + 2a - 2\(\sqrt{a + b}\)
Verified Correct Answer: A.
Step-by-Step Mathematical Solution
\(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\) = \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\) 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\(\sqrt{a(1 + a)}\)
Source: Myschool Past Questions Archive (Record #32609)