https://r2.erweima.ai/mathbot/image/1723167420120spudq8o3.jpg

The problem in the image is asking you to simplify the expression \(\frac{1}{\sqrt{2} - \sqrt{6}}\) such that there are no radicals in the denominator.

### Steps to Simplify:

1. **Multiply the numerator and denominator by the conjugate of the denominator:**
   The conjugate of \(\sqrt{2} - \sqrt{6}\) is \(\sqrt{2} + \sqrt{6}\).

   \[
   \frac{1}{\sqrt{2} - \sqrt{6}} \times \frac{\sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} = \frac{\sqrt{2} + \sqrt{6}}{(\sqrt{2} - \sqrt{6})(\sqrt{2} + \sqrt{6})}
   \]

2. **Simplify the denominator using the difference of squares formula:**
   \[
   (\sqrt{2} - \sqrt{6})(\sqrt{2} + \sqrt{6}) = (\sqrt{2})^2 - (\sqrt{6})^2 = 2 - 6 = -4
   \]

3. **Write the final simplified expression:**
   \[
   \frac{\sqrt{2} + \sqrt{6}}{-4} = -\frac{\sqrt{2} + \sqrt{6}}{4}
   \]

So, the expression \(\frac{1}{\sqrt{2} - \sqrt{6}}\) without radicals in the denominator is:

\[
-\frac{\sqrt{2} + \sqrt{6}}{4}
\]

Would you like further explanation on any of these steps, or do you have additional questions?

Here are 8 related questions you might consider:
1. What is the process of rationalizing the denominator?
2. How do you determine the conjugate of a binomial with radicals?
3. Can you apply the same process to expressions involving cube roots?
4. What is the difference of squares, and how is it used in simplification?
5. How would the expression change if the radicals were in the numerator instead?
6. What are some common pitfalls when rationalizing the denominator?
7. How would you simplify \(\frac{1}{\sqrt{a} - \sqrt{b}}\) for any \(a\) and \(b\)?
8. How does rationalizing the denominator help in real-world applications?

**Tip:** Rationalizing the denominator is a useful technique to simplify expressions and make them easier to work with, especially in more complex algebraic manipulations.

Learn how to simplify the expression \(\frac{1}{\sqrt{2} - \sqrt{6}}\) without radicals in the denominator. Follow our detailed step-by-step solution using conjugates and the difference of squares formula. Suitable for high school students studying algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation