What is the following quotient?

[tex]\[ \frac{5}{\sqrt{11} - \sqrt{3}} \][/tex]

A. [tex]\(\frac{5 \sqrt{11} - 5 \sqrt{3}}{8}\)[/tex]

B. [tex]\(\frac{5 \sqrt{11} + 5 \sqrt{3}}{8}\)[/tex]

C. [tex]\(\frac{5}{8}\)[/tex]

D. [tex]\(\frac{5 \sqrt{2}}{4}\)[/tex]



Answer :

To find the quotient [tex]\(\frac{5}{\sqrt{11} - \sqrt{3}}\)[/tex], we begin by rationalizing the denominator. The goal of rationalizing is to eliminate the square roots from the denominator.

Here are detailed steps to carry out the process:

1. Identify the Conjugate:
The conjugate of the denominator [tex]\(\sqrt{11} - \sqrt{3}\)[/tex] is [tex]\(\sqrt{11} + \sqrt{3}\)[/tex].

2. Multiply the Numerator and the Denominator by the Conjugate:
We need to multiply the numerator and denominator by [tex]\(\sqrt{11} + \sqrt{3}\)[/tex]:

[tex]\[ \frac{5}{\sqrt{11} - \sqrt{3}} \times \frac{\sqrt{11} + \sqrt{3}}{\sqrt{11} + \sqrt{3}} = \frac{5(\sqrt{11} + \sqrt{3})}{(\sqrt{11} - \sqrt{3})(\sqrt{11} + \sqrt{3})} \][/tex]

3. Simplify the Denominator:
Use the difference of squares formula, [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex]:

[tex]\[ (\sqrt{11})^2 - (\sqrt{3})^2 = 11 - 3 = 8 \][/tex]

So the denominator simplifies to 8.

4. Simplify the Numerator:
Distribute the 5 across [tex]\(\sqrt{11} + \sqrt{3}\)[/tex]:

[tex]\[ 5(\sqrt{11} + \sqrt{3}) = 5\sqrt{11} + 5\sqrt{3} \][/tex]

5. Combine the Numerator and Denominator:
The new, simplified fraction is:

[tex]\[ \frac{5\sqrt{11} + 5\sqrt{3}}{8} \][/tex]

Given the choices:

A) [tex]\(\frac{5 \sqrt{11} - 5 \sqrt{3}}{8}\)[/tex]

B) [tex]\(\frac{5 \sqrt{11} + 5 \sqrt{3}}{8}\)[/tex]

C) [tex]\(\frac{5}{8}\)[/tex]

D) [tex]\(\frac{5 \sqrt{2}}{4}\)[/tex]

We find that B) [tex]\(\frac{5\sqrt{11} + 5\sqrt{3}}{8}\)[/tex] is the correct answer.