Alright, let's solve the given problem step-by-step:
We start with the expression:
[tex]\[
\frac{1}{\sqrt{3}+1} + \frac{1}{\sqrt{3}-1}
\][/tex]
First, we need to find a common denominator for these two fractions. We can use the property of conjugate pairs to simplify this task. The common denominator we will use is [tex]\((\sqrt{3}+1)(\sqrt{3}-1)\)[/tex].
### Step 1: Simplify the Denominator
Using the difference of squares formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex], evaluate:
[tex]\[
(\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2
\][/tex]
So, the common denominator simplifies to 2.
### Step 2: Simplify Each Term
Next, express each original fraction with the common denominator:
#### For the first term:
[tex]\[
\frac{1}{\sqrt{3} + 1}
\][/tex]
Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((\sqrt{3} - 1)\)[/tex]:
[tex]\[
\frac{1 \cdot (\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{\sqrt{3} - 1}{2}
\][/tex]
#### For the second term:
[tex]\[
\frac{1}{\sqrt{3} - 1}
\][/tex]
Multiply the numerator and the denominator by the conjugate of the denominator [tex]\((\sqrt{3} + 1)\)[/tex]:
[tex]\[
\frac{1 \cdot (\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{\sqrt{3} + 1}{2}
\][/tex]
### Step 3: Combine the Fractions
Now, we add these two fractions with the common denominator:
[tex]\[
\frac{\sqrt{3} - 1}{2} + \frac{\sqrt{3} + 1}{2}
\][/tex]
Combine the numerators:
[tex]\[
\frac{\sqrt{3} - 1 + \sqrt{3} + 1}{2} = \frac{2\sqrt{3}}{2}
\][/tex]
Simplifying the numerator:
[tex]\[
\frac{2\sqrt{3}}{2} = \sqrt{3}
\][/tex]
### Conclusion
Therefore, the sum of the fractions is:
[tex]\[
\frac{1}{\sqrt{3}+1} + \frac{1}{\sqrt{3}-1} = \sqrt{3}
\][/tex]
