To solve the expression \( \frac{10}{\sqrt{5} + \sqrt{2}} \), we should first rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

### Step 1: Identify the conjugate
The conjugate of \( \sqrt{5} + \sqrt{2} \) is \( \sqrt{5} - \sqrt{2} \).

### Step 2: Multiply the numerator and denominator by the conjugate
\[
\frac{10}{\sqrt{5} + \sqrt{2}} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{10(\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})}
\]

### Step 3: Simplify the denominator
Using the difference of squares formula, \( (a + b)(a - b) = a^2 - b^2 \), we get:
\[
(\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3
\]

### Step 4: Simplify the expression
Now the expression becomes:
\[
\frac{10(\sqrt{5} - \sqrt{2})}{3} = \frac{10\sqrt{5} - 10\sqrt{2}}{3} = \frac{10\sqrt{5}}{3} - \frac{10\sqrt{2}}{3}
\]

So, the simplified form of the expression is:
\[
\frac{10\sqrt{5}}{3} - \frac{10\sqrt{2}}{3}
\]

Would you like any further details or explanations?

Here are some related questions you might find helpful:
1. How do you rationalize the denominator of a fraction?
2. What is the difference of squares formula, and how is it applied?
3. How do you simplify expressions involving square roots?
4. What is the conjugate of a binomial expression, and why is it useful?
5. How can you simplify complex fractions?

**Tip:** When rationalizing the denominator, always multiply by the conjugate to remove the square roots from the denominator.

Learn how to simplify the expression \( \frac{10}{\sqrt{5} + \sqrt{2}} \) step-by-step using the conjugate of the denominator. This problem involves rationalizing the denominator and simplifying fractions, suitable for students studying advanced algebra concepts.

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