https://file.aiquickdraw.com/mathbot/file/1724673410710ehlhdc4a.jpg

The expression in the image appears to be:

\[
\frac{5 \cdot \sqrt{a \times n}}{\sqrt{a^n}}
\]

To simplify this expression, follow these steps:

### Step 1: Simplify the square root terms.
The expression inside the square roots can be rewritten as:

\[
\frac{5 \cdot \sqrt{a \cdot n}}{\sqrt{a^n}} = 5 \cdot \frac{\sqrt{a \cdot n}}{\sqrt{a^n}}
\]

### Step 2: Simplify the fraction of square roots.
Use the property of square roots:

\[
\frac{\sqrt{a \cdot n}}{\sqrt{a^n}} = \sqrt{\frac{a \cdot n}{a^n}}
\]

### Step 3: Simplify the expression inside the square root.
\[
\frac{a \cdot n}{a^n} = \frac{n}{a^{n-1}}
\]

### Step 4: Substitute back and simplify.
Now the expression becomes:

\[
5 \cdot \sqrt{\frac{n}{a^{n-1}}}
\]

This is the simplified form of the given expression.

Would you like any additional details or explanations?

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Here are five related questions that expand on this topic:
1. How can this expression be simplified if \( n \) is a specific integer?
2. What if \( n = 2 \)? How would the expression change?
3. How does the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) apply to this problem?
4. How can this expression be further simplified if \( a \) and \( n \) are both known values?
5. How would the expression look if instead of \( \sqrt{a \cdot n} \) it was \( \sqrt{a + n} \)?

**Tip:** Remember that when dealing with square roots, simplifying the terms inside the root first can make the overall expression much easier to manage.

Learn how to simplify the expression \( \frac{5 \cdot \sqrt{a \times n}}{\sqrt{a^n}} \) step-by-step with clear explanations and examples. This problem involves square roots and fraction simplification, suitable for high school students and beyond.

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