Solve the following :
If `x^(x^(3/2)) = (x^(3/2))^x`, then the value of x is :

Options

A. `2/3`
B. `9/4`
C. `16/25`
D. `8/27`

Answer

To solve the equation \( x^{(x^{3/2})} = (x^{3/2})^x \), we can follow these steps:

### Step 1: Rewrite the right-hand side using the power of a power property
The expression on the right-hand side can be rewritten using the property of exponents \( (a^m)^n = a^{m \cdot n} \). Thus, we have:
\[ (x^{3/2})^x = x^{(3/2) \cdot x} \]
So, the equation becomes:
\[ x^{(x^{3/2})} = x^{(3/2) \cdot x} \]

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

### Step 3: Rearrange the equation
To eliminate the fraction, we can multiply both sides by 2:
\[ 2 \cdot x^{3/2} = 3x \]

### Step 4: Rearranging to isolate terms
Rearranging gives us:
\[ 2x^{3/2} - 3x = 0 \]

### Step 5: Factor out common terms
We can factor out \( x \):
\[ x(2x^{1/2} - 3) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x = 0 \) (not valid since \( x \) must be positive in this context)
2. \( 2x^{1/2} - 3 = 0 \)

Solving the second equation:
\[ 2x^{1/2} = 3 \implies x^{1/2} = \frac{3}{2} \]
Squaring both sides:
\[ x = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

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

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