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

Options

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

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ x^{x\sqrt{x}} = (x\sqrt{x})^x \]

### Step 2: Simplify the right-hand side
The right-hand side can be rewritten as:
\[ (x\sqrt{x})^x = (x^{1.5})^x = x^{1.5x} \]
So, we can now rewrite the equation as:
\[ x^{x\sqrt{x}} = x^{1.5x} \]

### Step 3: Set the exponents equal
Since the bases are the same (both are \( x \)), we can set the exponents equal to each other:
\[ x\sqrt{x} = 1.5x \]

### Step 4: Divide both sides by \( x \) (assuming \( x \neq 0 \))
We can divide both sides by \( x \):
\[ \sqrt{x} = 1.5 \]

### Step 5: Square both sides
Next, we square both sides to eliminate the square root:
\[ x = (1.5)^2 \]

### Step 6: Calculate \( (1.5)^2 \)
Calculating \( (1.5)^2 \):
\[ (1.5)^2 = 2.25 \]
Thus, we have:
\[ x = 2.25 \]

### Step 7: Convert to fraction
To express \( 2.25 \) as a fraction:
\[ 2.25 = \frac{9}{4} \]

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