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.
