Join IDNLearn.com and start exploring the answers to your most pressing questions. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.

What is the following quotient?

[tex]\[
\frac{1}{1+\sqrt{3}}
\][/tex]

A. [tex]\(\frac{\sqrt{3}}{4}\)[/tex]

B. [tex]\(\frac{1+\sqrt{3}}{4}\)[/tex]

C. [tex]\(\frac{1-\sqrt{3}}{4}\)[/tex]

D. [tex]\(\frac{-1+\sqrt{3}}{2}\)[/tex]


Sagot :

To determine the quotient of [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex], we need to rationalize the denominator, a process that eliminates the square root from the denominator.

### Step 1: Rationalize the denominator
To rationalize the denominator of [tex]\(\frac{1}{1 + \sqrt{3}}\)[/tex], multiply both the numerator and the denominator by the conjugate of the denominator, [tex]\(1 - \sqrt{3}\)[/tex]. The conjugate is used because it can simplify the expressions involving square roots.

[tex]\[ \frac{1}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{1 \cdot (1 - \sqrt{3})}{(1 + \sqrt{3}) \cdot (1 - \sqrt{3})} \][/tex]

### Step 2: Simplify the numerator
The numerator becomes:

[tex]\[ 1 \cdot (1 - \sqrt{3}) = 1 - \sqrt{3} \][/tex]

### Step 3: Simplify the denominator
Use the difference of squares formula for the denominator:

[tex]\[ (1 + \sqrt{3})(1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \][/tex]

So, the expression becomes:

[tex]\[ \frac{1 - \sqrt{3}}{-2} \][/tex]

### Step 4: Remove the negative from the denominator
Simplify by changing the signs in the numerator:

[tex]\[ \frac{1 - \sqrt{3}}{-2} = \frac{-(1 - \sqrt{3})}{2} = \frac{-1 + \sqrt{3}}{2} \][/tex]

### Step 5: Compare to given choices
The answer matches one of the given choices. Among the choices given:
1. [tex]\(\frac{\sqrt{3}}{4}\)[/tex]
2. [tex]\(\frac{1 + \sqrt{3}}{4}\)[/tex]
3. [tex]\(\frac{1 - \sqrt{3}}{4}\)[/tex]
4. [tex]\(\frac{-1 + \sqrt{3}}{2}\)[/tex]

The correct choice is:

[tex]\[ \boxed{\frac{-1 + \sqrt{3}}{2}} \][/tex]