https://file.aiquickdraw.com/mathbot/file/1727247556603jbsneyej.jpg

The expression in the image is:

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

To simplify this, we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{10} + \sqrt{5}\) is \(\sqrt{10} - \sqrt{5}\).

Let's go step by step:

1. Multiply the numerator and the denominator by \(\sqrt{10} - \sqrt{5}\):

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

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

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

So, the expression becomes:

\[
\frac{5(\sqrt{10} - \sqrt{5})}{5}
\]

3. The \(5\)s cancel out, and we are left with:

\[
\sqrt{10} - \sqrt{5}
\]

Thus, the simplified expression is:

\[
\sqrt{10} - \sqrt{5}
\]

Would you like further clarification on any part? Here are some related questions for practice:

1. How do you rationalize a denominator involving a sum of square roots?
2. What is the difference between a sum and difference of squares in terms of simplification?
3. Can you simplify \(\frac{7}{\sqrt{3} + \sqrt{2}}\) using the same method?
4. What are conjugates and why are they useful in rationalizing denominators?
5. What happens if you have a complex number in the denominator? How do you rationalize it?

**Tip:** Rationalizing the denominator helps avoid square roots or irrational numbers in the denominator, making expressions easier to work with.

Simplify the expression \( \frac{5}{\sqrt{10} + \sqrt{5}} \)

Learn how to simplify the expression \( \frac{5}{\sqrt{10} + \sqrt{5}} \) by rationalizing the denominator. This solution involves algebraic concepts like the difference of squares theorem and rationalization of square roots. Suitable for students in Grades 9-12.

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