If `x^(xsqrt(x))=(xsqrt(x))^(x)`, then x is equal to:

Options

A. `3/2`
B. `2/9`
C. `9/4`
D. `4/9`

Answer

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

### Step 1: Rewrite the Right-Hand Side (RHS)
The RHS is \( (x\sqrt{x})^{x} \). We can rewrite \( \sqrt{x} \) as \( x^{1/2} \). Therefore, we have:
\[ x\sqrt{x} = x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2} \]
So, the RHS becomes:
\[ (x\sqrt{x})^{x} = (x^{3/2})^{x} \]

### Step 2: Apply the Power of a Power Rule
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we can simplify the RHS:
\[ (x^{3/2})^{x} = x^{(3/2) \cdot x} = x^{(3x/2)} \]

### Step 3: Set the Left-Hand Side (LHS) Equal to the RHS
Now, we can set the LHS equal to the RHS:
\[ x^{x\sqrt{x}} = x^{(3x/2)} \]

### Step 4: Equate the Exponents
Since the bases are the same (both are \( x \)), we can equate the exponents:
\[ x\sqrt{x} = \frac{3x}{2} \]

### Step 5: Simplify the Equation
We can divide both sides by \( x \) (assuming \( x \neq 0 \)):
\[ \sqrt{x} = \frac{3}{2} \]

### Step 6: Square Both Sides
To eliminate the square root, we square both sides:
\[ (\sqrt{x})^2 = \left(\frac{3}{2}\right)^2 \]
This simplifies to:
\[ x = \frac{9}{4} \]

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

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