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The given expression is:

\[
\frac{5 \sqrt{2}}{\sqrt{10} - \sqrt{5}} - \frac{3 \sqrt{5}}{\sqrt{5} - \sqrt{2}} - \frac{\sqrt{10}}{\sqrt{2} - 1}
\]

To simplify this, we need to rationalize the denominators of each fraction. Let's handle each term individually.

### Step 1: Rationalize \(\frac{5 \sqrt{2}}{\sqrt{10} - \sqrt{5}}\)
We multiply both the numerator and denominator by the conjugate of the denominator, \(\sqrt{10} + \sqrt{5}\).

\[
\frac{5 \sqrt{2}}{\sqrt{10} - \sqrt{5}} \cdot \frac{\sqrt{10} + \sqrt{5}}{\sqrt{10} + \sqrt{5}} = \frac{5 \sqrt{2}(\sqrt{10} + \sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}
\]

The denominator simplifies to:

\[
(\sqrt{10})^2 - (\sqrt{5})^2 = 10 - 5 = 5
\]

Now the expression becomes:

\[
\frac{5 \sqrt{2}(\sqrt{10} + \sqrt{5})}{5} = \sqrt{2}(\sqrt{10} + \sqrt{5})
\]

Distribute the \(\sqrt{2}\):

\[
\sqrt{2} \cdot \sqrt{10} = \sqrt{20} = 2 \sqrt{5}, \quad \sqrt{2} \cdot \sqrt{5} = \sqrt{10}
\]

So, the first term becomes:

\[
2 \sqrt{5} + \sqrt{10}
\]

### Step 2: Rationalize \(\frac{3 \sqrt{5}}{\sqrt{5} - \sqrt{2}}\)
We multiply both the numerator and denominator by the conjugate of the denominator, \(\sqrt{5} + \sqrt{2}\):

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

The denominator simplifies to:

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

Now the expression becomes:

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

Distribute the \(\sqrt{5}\):

\[
\sqrt{5} \cdot \sqrt{5} = 5, \quad \sqrt{5} \cdot \sqrt{2} = \sqrt{10}
\]

So, the second term becomes:

\[
5 + \sqrt{10}
\]

### Step 3: Rationalize \(\frac{\sqrt{10}}{\sqrt{2} - 1}\)
We multiply both the numerator and denominator by the conjugate of the denominator, \(\sqrt{2} + 1\):

\[
\frac{\sqrt{10}}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{\sqrt{10}(\sqrt{2} + 1)}{(\sqrt{2})^2 - 1^2}
\]

The denominator simplifies to:

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

So, the third term becomes:

\[
\sqrt{10}(\sqrt{2} + 1) = \sqrt{10} \cdot \sqrt{2} + \sqrt{10} = \sqrt{20} + \sqrt{10} = 2 \sqrt{5} + \sqrt{10}
\]

### Step 4: Combine the terms
We now have:

\[
(2 \sqrt{5} + \sqrt{10}) - (5 + \sqrt{10}) - (2 \sqrt{5} + \sqrt{10})
\]

Simplify by combining like terms:

1. The \(\sqrt{10}\) terms cancel out since \(+\sqrt{10} - \sqrt{10} - \sqrt{10} = 0\).
2. The \(2 \sqrt{5}\) terms cancel out as well: \(2 \sqrt{5} - 2 \sqrt{5} = 0\).

This leaves:

\[
0 - 5 = -5
\]

Thus, the simplified expression is:

\[
\boxed{-5}
\]

Would you like more details on any step? Here are some related questions to expand upon:

1. What are conjugates, and how are they useful in rationalizing denominators?
2. Why do we rationalize the denominator, and how does it affect expressions?
3. Can you show examples of simplifying expressions with other radicals?
4. How does one simplify the product of two square roots?
5. How do you approach more complex radical expressions?

**Tip:** Always look for common terms that can cancel out when simplifying radical expressions to make calculations easier.

Simplify the expression \( \frac{5\sqrt{2}}{\sqrt{10}-\sqrt{5}} - \frac{3\sqrt{5}}{\sqrt{5}-\sqrt{2}} - \frac{\sqrt{10}}{\sqrt{2}-1} \).

Learn how to simplify the expression \( \frac{5\sqrt{2}}{\sqrt{10}-\sqrt{5}} - \frac{3\sqrt{5}}{\sqrt{5}-\sqrt{2}} - \frac{\sqrt{10}}{\sqrt{2}-1} \) by using rationalization techniques. This involves key concepts like multiplying by conjugates and simplifying radical expressions. Suitable for students in Grades 10-12.

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