Question

what is the following quotient? \\(\frac{5}{\sqrt{11}-\sqrt{3}}\\) \\(\frac{5\sqrt{11}+5\sqrt{3}}{8}\\) \\(\frac{5}{8}\\) \\(\frac{5\sqrt{11}-5\sqrt{3}}{8}\\) \\(\frac{5\sqrt{2}}{4}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of \(\frac{5}{\sqrt{11}-\sqrt{3}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{11}+\sqrt{3}\). So we have:

$$ \frac{5(\sqrt{11}+\sqrt{3})}{(\sqrt{11}-\sqrt{3})(\sqrt{11}+\sqrt{3})} $$

Step2: Simplify the denominator

Using the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a = \sqrt{11}\) and \(b=\sqrt{3}\), the denominator becomes \((\sqrt{11})^{2}-(\sqrt{3})^{2}=11 - 3=8\).

Step3: Simplify the numerator

The numerator is \(5(\sqrt{11}+\sqrt{3})=5\sqrt{11}+5\sqrt{3}\).

Step4: Write the final fraction

Putting it all together, we get \(\frac{5\sqrt{11}+5\sqrt{3}}{8}\).

Answer:

\(\frac{5\sqrt{11}+5\sqrt{3}}{8}\) (corresponding to the option with this expression)