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.