Answered

Get the information you need with the help of IDNLearn.com's expert community. Join our interactive community and get comprehensive, reliable answers to all your questions.

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]

Sagot :

GinnyAnsweridn
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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Accurate answers are just a click away at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.