https://file.aiquickdraw.com/mathbot/file/1729532063852xrowdqwz.jpg

The expression provided in the image is \( x \sqrt{x^7} \), and the task is to simplify it to its simplest radical form.

### Step-by-step solution:

1. **Simplifying the radical**:
   \[
   \sqrt{x^7} = \sqrt{x^6 \cdot x} = \sqrt{(x^3)^2 \cdot x}
   \]
   Since \( \sqrt{(x^3)^2} = x^3 \), we get:
   \[
   \sqrt{x^7} = x^3 \sqrt{x}
   \]

2. **Multiplying by the outside term**:
   Now, we multiply this result by the original \( x \):
   \[
   x \cdot x^3 \sqrt{x} = x^4 \sqrt{x}
   \]

Thus, the expression simplifies to \( x^4 \sqrt{x} \).

### Correct answer:
The correct answer is **\( x^4 \sqrt{x} \)**, which matches option (C) in the image.

Would you like further details on this solution or have any questions?

Here are 5 related questions for deeper understanding:
1. How do you simplify expressions with square roots involving exponents?
2. Why can we factor powers of \( x \) inside square roots?
3. How would the simplification differ if it were a cube root instead of a square root?
4. Can you simplify \( \sqrt{x^{10}} \) in a similar way?
5. How do you handle mixed radical and exponential expressions in general?

**Tip**: Always look for perfect squares when simplifying square roots involving variables to simplify the expression more easily.

Express x√(x^7) in simplest radical form.

Learn how to simplify the radical expression x√(x^7) into its simplest form. This problem covers key concepts like radicals and exponents, and provides a detailed step-by-step solution. Suitable for high school students in Grades 9-12.

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