If `x^(x(sqrtx)) = (xsqrt(x))^(x)`, then the value of x is:

Options

A. `9/4`
B. `4/9`
C. 8
D. 4

Answer

To solve the equation \( x^{x\sqrt{x}} = (x\sqrt{x})^x \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ x^{x\sqrt{x}} = (x\sqrt{x})^x \]
We can express \(\sqrt{x}\) as \(x^{1/2}\):
\[ x^{x \cdot x^{1/2}} = (x \cdot x^{1/2})^x \]

### Step 2: Simplify the right-hand side
The right-hand side can be simplified:
\[ (x \cdot x^{1/2})^x = (x^{1 + 1/2})^x = (x^{3/2})^x \]
This simplifies to:
\[ x^{(3/2)x} \]

### Step 3: Set the exponents equal
Now we have:
\[ x^{x \cdot x^{1/2}} = x^{(3/2)x} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x \cdot x^{1/2} = \frac{3}{2}x \]

### Step 4: Simplify the left-hand side
The left-hand side can be rewritten as:
\[ x^{1 + 1/2} = x^{3/2} \]
So we have:
\[ x^{3/2} = \frac{3}{2}x \]

### Step 5: Divide both sides by \(x\) (assuming \(x \neq 0\))
Dividing both sides by \(x\) gives:
\[ x^{1/2} = \frac{3}{2} \]

### Step 6: Square both sides
Now we square both sides to eliminate the square root:
\[ (x^{1/2})^2 = \left(\frac{3}{2}\right)^2 \]
This results in:
\[ x = \frac{9}{4} \]

### Conclusion
The value of \(x\) is:
\[ \boxed{\frac{9}{4}} \]