https://r2.erweima.ai/mathbot/image/1726434486971gwrqtylj.jpg

The expression in the image is:

\[
\frac{\sqrt{3} + 1}{\sqrt{3} - 1}
\]

We can simplify this expression by rationalizing the denominator. To do this, we'll multiply both the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{3} + 1\).

### Step-by-Step Process:

1. Multiply both numerator and denominator by \(\sqrt{3} + 1\):
\[
\frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}
\]

2. Use the difference of squares formula on the denominator:
\[
(\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2
\]

3. Expand the numerator:
\[
(\sqrt{3} + 1)(\sqrt{3} + 1) = (\sqrt{3})^2 + 2 \cdot \sqrt{3} \cdot 1 + 1^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}
\]

4. Now the expression becomes:
\[
\frac{4 + 2\sqrt{3}}{2}
\]

5. Simplify the numerator:
\[
\frac{4}{2} + \frac{2\sqrt{3}}{2} = 2 + \sqrt{3}
\]

Thus, the simplified form of the given expression is:

\[
2 + \sqrt{3}
\]

Let me know if you'd like further details on any step or have additional questions! Here are some related questions:

1. How do you rationalize the denominator in general?
2. What is the difference of squares formula and why is it used here?
3. Can the expression \(\frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) be simplified in other ways?
4. What happens if you multiply both numerator and denominator by a different number?
5. What are some practical applications of simplifying such expressions?

**Tip:** Always check if you can simplify an expression by rationalizing the denominator to make it more usable in calculations.

Simplify the expression \(\frac{\sqrt{3} + 1}{\sqrt{3} - 1}\).

Learn how to simplify the expression (√3 + 1) / (√3 - 1) by rationalizing the denominator. This tutorial covers key algebraic concepts and the difference of squares formula, providing a step-by-step solution for students in Grades 9-12.

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