https://file.aiquickdraw.com/mathbot/file/1726621186702tomcrxse.jpg

The expression shown in the image is:

\[
\frac{5}{3\sqrt{5} + \sqrt{2}}
\]

To simplify this, we will rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, \( 3\sqrt{5} - \sqrt{2} \).

Let's go through the steps:

1. **Multiply the numerator and denominator** by \( 3\sqrt{5} - \sqrt{2} \):

\[
\frac{5}{3\sqrt{5} + \sqrt{2}} \times \frac{3\sqrt{5} - \sqrt{2}}{3\sqrt{5} - \sqrt{2}} = \frac{5(3\sqrt{5} - \sqrt{2})}{(3\sqrt{5})^2 - (\sqrt{2})^2}
\]

2. **Simplify the denominator** using the difference of squares formula:

\[
(3\sqrt{5})^2 = 9 \times 5 = 45 \quad \text{and} \quad (\sqrt{2})^2 = 2
\]
So the denominator becomes:
\[
45 - 2 = 43
\]

3. **Simplify the numerator**:

\[
5(3\sqrt{5} - \sqrt{2}) = 15\sqrt{5} - 5\sqrt{2}
\]

4. **Final expression**:

\[
\frac{15\sqrt{5} - 5\sqrt{2}}{43}
\]

Thus, the simplified form of the expression is:

\[
\frac{15\sqrt{5} - 5\sqrt{2}}{43}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you rationalize more complex expressions?
2. What are the properties of square roots used in this example?
3. Can you generalize the process of rationalizing denominators with square roots?
4. What are other ways to simplify fractional expressions with roots?
5. How does the difference of squares formula work in algebraic simplifications?

**Tip:** Rationalizing a denominator helps in making expressions easier to evaluate, especially when dealing with roots.

Simplify the expression 5 / (3√5 + √2)

Learn how to rationalize and simplify the expression 5 / (3√5 + √2). This example demonstrates rationalizing the denominator and applying the difference of squares theorem to obtain the final simplified form. Suitable for students in Grades 10-12.

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