To solve for the quotient [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex], we need to rationalize the denominator. Here is a step-by-step explanation of the process.
1. Original Expression:
[tex]\[
\frac{1}{1 + \sqrt{3}}
\][/tex]
2. Rationalize the Denominator:
To get rid of the square root in the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(1 + \sqrt{3}\)[/tex] is [tex]\(1 - \sqrt{3}\)[/tex].
[tex]\[
\frac{1}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{1 \cdot (1 - \sqrt{3})}{(1 + \sqrt{3}) \cdot (1 - \sqrt{3})}
\][/tex]
3. Multiply the Numerator:
[tex]\[
1 \cdot (1 - \sqrt{3}) = 1 - \sqrt{3}
\][/tex]
4. Multiply the Denominator:
The denominator is a difference of squares:
[tex]\[
(1 + \sqrt{3})(1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2
\][/tex]
5. Form the New Fraction:
[tex]\[
\frac{1 - \sqrt{3}}{-2}
\][/tex]
6. Simplify the Fraction:
Divide both terms in the numerator by [tex]\(-2\)[/tex]:
[tex]\[
\frac{1 - \sqrt{3}}{-2} = \frac{1}{-2} - \frac{\sqrt{3}}{-2} = -\frac{1}{2} + \frac{\sqrt{3}}{2}
\][/tex]
7. Combine into a Single Fraction:
[tex]\[
= \frac{\sqrt{3} - 1}{2}
\][/tex]
So, the simplified quotient is:
[tex]\[
\frac{\sqrt{3} - 1}{2}
\][/tex]
Comparing this result with the given options:
- [tex]\(\frac{\sqrt{3}}{4}\)[/tex]
- [tex]\(\frac{1+\sqrt{3}}{4}\)[/tex]
- [tex]\(\frac{1-\sqrt{3}}{4}\)[/tex]
- [tex]\(\frac{-1+\sqrt{3}}{2}\)[/tex]
The correct option is:
[tex]\[\frac{-1+\sqrt{3}}{2}\][/tex]
Therefore, the quotient [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex] simplified is:
[tex]\[
\boxed{\frac{-1+\sqrt{3}}{2}}
\][/tex]
