The expression \( \frac{2}{\sqrt{2} + \sqrt{3}} \) can be simplified by rationalizing the denominator. Here’s how to do it:

### Step 1: Multiply both numerator and denominator by the conjugate of the denominator.
The conjugate of \( \sqrt{2} + \sqrt{3} \) is \( \sqrt{2} - \sqrt{3} \). So, multiply both the numerator and the denominator by \( \sqrt{2} - \sqrt{3} \):

\[
\frac{2}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} = \frac{2(\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})}
\]

### Step 2: Simplify the denominator.
The denominator is now in the form \( (a+b)(a-b) \), which simplifies as:

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

### Step 3: Final expression.
Now simplify the entire expression:

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

So, the simplified expression is:

\[
-2(\sqrt{2} - \sqrt{3}) = -2\sqrt{2} + 2\sqrt{3}
\]

Thus, \( \frac{2}{\sqrt{2} + \sqrt{3}} = -2\sqrt{2} + 2\sqrt{3} \).

Would you like further details on this, or do you have any questions?

### 5 Related Questions:
1. How do you rationalize the denominator for more complex expressions?
2. What is the geometric meaning of conjugates in radicals?
3. Can you explain more on how the difference of squares formula works?
4. How do you handle expressions with cube roots?
5. What happens when you rationalize a denominator with complex numbers?

### Tip:
When rationalizing the denominator, always multiply by the conjugate of the denominator, which helps eliminate the square roots.

Learn how to simplify the expression 2 / (√2 + √3) by rationalizing the denominator. This problem involves key algebraic concepts such as radicals and the difference of squares formula. A step-by-step solution is provided, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation