https://file.aiquickdraw.com/mathbot/file/1725388950116wxjxu157.jpg

The expression in the image appears to be:

\[
\frac{2}{\sqrt{5}} - \frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10} + \sqrt{6}}{\sqrt{6}}
\]

To simplify this expression, we'll go step by step:

### Step 1: Simplify the product involving radicals.
The expression inside the product is:

\[
\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10} + \sqrt{6}}{\sqrt{6}}
\]

Multiply the fractions:

\[
= \frac{1 \cdot (\sqrt{10} + \sqrt{6})}{\sqrt{10} \cdot \sqrt{6}}
\]

Simplifying the denominator:

\[
\sqrt{10} \cdot \sqrt{6} = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}
\]

So, the expression becomes:

\[
\frac{\sqrt{10} + \sqrt{6}}{2\sqrt{15}}
\]

### Step 2: Combine the expressions with a common denominator.
Now, the original expression is:

\[
\frac{2}{\sqrt{5}} - \frac{\sqrt{10} + \sqrt{6}}{2\sqrt{15}}
\]

To combine these, we'll first rationalize the denominators.

### Step 3: Rationalize the denominators.
- For \(\frac{2}{\sqrt{5}}\), multiply numerator and denominator by \(\sqrt{5}\):

\[
\frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
\]

- For \(\frac{\sqrt{10} + \sqrt{6}}{2\sqrt{15}}\), multiply numerator and denominator by \(\sqrt{15}\):

\[
\frac{\sqrt{10} + \sqrt{6}}{2\sqrt{15}} = \frac{(\sqrt{10} + \sqrt{6})\sqrt{15}}{30}
\]

This gives us:

\[
\frac{2\sqrt{5}}{5} - \frac{\sqrt{150} + \sqrt{90}}{30}
\]

### Step 4: Simplify the radicals and combine.
Simplify the radicals in the second term:

\[
\sqrt{150} = \sqrt{25 \times 6} = 5\sqrt{6}
\]
\[
\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}
\]

So the expression becomes:

\[
\frac{2\sqrt{5}}{5} - \frac{5\sqrt{6} + 3\sqrt{10}}{30}
\]

Now, combine the fractions by finding a common denominator:

The common denominator between 5 and 30 is 30, so:

\[
\frac{2\sqrt{5} \times 6}{30} - \frac{5\sqrt{6} + 3\sqrt{10}}{30}
\]

This simplifies to:

\[
\frac{12\sqrt{5} - (5\sqrt{6} + 3\sqrt{10})}{30}
\]

Finally, distribute the negative sign:

\[
\frac{12\sqrt{5} - 5\sqrt{6} - 3\sqrt{10}}{30}
\]

### Final Answer:
The simplified expression is:

\[
\frac{12\sqrt{5} - 5\sqrt{6} - 3\sqrt{10}}{30}
\]

This is the final simplified form.

Would you like more details or have any questions?

### Related Questions:
1. How do you rationalize denominators in fractions involving square roots?
2. What are some methods to simplify complex fractions?
3. Can you explain how to find a common denominator in algebraic expressions?
4. How do you handle radicals when adding or subtracting fractions?
5. What steps should be taken when multiplying two fractions with square roots?

### Tip:
When simplifying expressions involving square roots, always look for opportunities to rationalize the denominator early in the process to make combining terms easier.

Simplify the expression involving square roots: \(\frac{2}{\sqrt{5}} - \frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10} + \sqrt{6}}{\sqrt{6}}\)

This detailed solution simplifies the expression \(\frac{2}{\sqrt{5}} - \frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10} + \sqrt{6}}{\sqrt{6}}\) by rationalizing the denominators and combining like terms. Ideal for students in Grades 10-12.

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