What is the value of \(\displaystyle\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}+\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
Add the two fractions over the common denominator \((\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})\). By \((a-b)(a+b)=a^{2}-b^{2}\), this denominator is a whole number, and the root terms in the numerator cancel.
Denominator: \((\sqrt{7})^{2}-(\sqrt{3})^{2}=7-3=4\)
\((\sqrt{7}+\sqrt{3})^{2}=7+3+2\sqrt{21}=10+2\sqrt{21}\)
\((\sqrt{7}-\sqrt{3})^{2}=7+3-2\sqrt{21}=10-2\sqrt{21}\)
Numerator: \((10+2\sqrt{21})+(10-2\sqrt{21})=20\)
\(\displaystyle\frac{20}{4}=5\)