If `f(x)=sqrt(x^(3))`, then f (3x) will be equal to

Options

A. `sqrt(3x^(3))`
B. `3sqrt(x^(3))`
C. `3 sqrt((3x^(3)))`
D. `3sqrt(x^(5))`

Answer

To solve the problem, we need to find \( f(3x) \) given that \( f(x) = \sqrt{x^3} \).

### Step-by-Step Solution:

1. **Write down the function**:
We start with the given function:
\[ f(x) = \sqrt{x^3} \]

2. **Substitute \( 3x \) into the function**:
We need to find \( f(3x) \). To do this, we replace \( x \) with \( 3x \):
\[ f(3x) = \sqrt{(3x)^3} \]

3. **Simplify \( (3x)^3 \)**:
Next, we calculate \( (3x)^3 \):
\[ (3x)^3 = 3^3 \cdot x^3 = 27x^3 \]
Therefore, we have:
\[ f(3x) = \sqrt{27x^3} \]

4. **Factor out the square root**:
We can simplify \( \sqrt{27x^3} \) further. We know that:
\[ 27 = 9 \cdot 3 = 3^2 \cdot 3 \]
Thus, we can write:
\[ \sqrt{27x^3} = \sqrt{9 \cdot 3 \cdot x^3} = \sqrt{9} \cdot \sqrt{3} \cdot \sqrt{x^3} \]

5. **Calculate the square roots**:
Now we calculate the square roots:
\[ \sqrt{9} = 3 \quad \text{and} \quad \sqrt{x^3} = x^{3/2} \]
Therefore, we have:
\[ f(3x) = 3 \cdot \sqrt{3} \cdot x^{3/2} \]

6. **Final expression**:
We can express the final result as:
\[ f(3x) = 3\sqrt{3} \cdot x^{3/2} \]

### Conclusion:
The expression for \( f(3x) \) is:
\[ f(3x) = 3\sqrt{3} \cdot x^{3/2} \]