What is the value of \(\displaystyle\frac{1+a}{a^{\frac{1}{2}}+a^{-\frac{1}{2}}}-\frac{a^{\frac{1}{2}}+a^{-\frac{1}{2}}}{1+a}+a^{-\frac{1}{2}}\), where \(a>0\)?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\sqrt{a}\)
Write \(s=\sqrt{a}\), so that \(a^{\frac{1}{2}}=s\) and \(a^{-\frac{1}{2}}=\frac{1}{s}\). Then \(s+\frac{1}{s}=\frac{s^2+1}{s}=\frac{1+a}{s}\), which is the piece that links the two fractions.
First fraction: \(\displaystyle\frac{1+a}{\frac{1+a}{s}}=s\)
Second fraction: \(\displaystyle\frac{\frac{1+a}{s}}{1+a}=\frac{1}{s}\)
Whole expression: \(s-\displaystyle\frac{1}{s}+\frac{1}{s}=s=\sqrt{a}\)