Let `f(x)=sqrt(x^(5))`, then `f(5x)` is equal to :

Options

A. `5xsqrt(5x)`
B. `25x^(2)sqrt(5x)`
C. `5x`
D. `sqrt(5x^(2))`

Answer

To solve the problem, we need to find the value of \( f(5x) \) given that \( f(x) = \sqrt{x^5} \).

### Step-by-Step Solution:

1. **Identify the function**: We are given the function \( f(x) = \sqrt{x^5} \).

2. **Substitute \( 5x \) into the function**: To find \( f(5x) \), we replace \( x \) in the function with \( 5x \):
\[ f(5x) = \sqrt{(5x)^5} \]

3. **Simplify \( (5x)^5 \)**: We can expand \( (5x)^5 \) using the property of exponents:
\[ (5x)^5 = 5^5 \cdot x^5 \]

4. **Substitute back into the square root**: Now, we substitute this back into the square root:
\[ f(5x) = \sqrt{5^5 \cdot x^5} \]

5. **Use the property of square roots**: The square root of a product can be expressed as the product of the square roots:
\[ f(5x) = \sqrt{5^5} \cdot \sqrt{x^5} \]

6. **Calculate \( \sqrt{5^5} \) and \( \sqrt{x^5} \)**:
- For \( \sqrt{5^5} \), we can simplify it:
\[ \sqrt{5^5} = 5^{5/2} = 5^2 \cdot \sqrt{5} = 25\sqrt{5} \]
- For \( \sqrt{x^5} \), we similarly get:
\[ \sqrt{x^5} = x^{5/2} = x^2 \cdot \sqrt{x} \]

7. **Combine the results**: Now we can combine these results:
\[ f(5x) = 25\sqrt{5} \cdot x^2 \cdot \sqrt{x} = 25x^2\sqrt{5x} \]

### Final Answer:
Thus, the value of \( f(5x) \) is:
\[ f(5x) = 25x^2\sqrt{5x} \]