If `x^(x^(3/2))=(x^(3/2))^x`, the value of x :

Options

A. a)`4/9`
B. b)`9/4`
C. c)`3/2`
D. d)`2/3`

Answer

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

### Step 1: Rewrite the equation
The given equation is:
\[ x^{x^{3/2}} = (x^{3/2})^x \]
We can rewrite the right side using the property of exponents:
\[ (x^{3/2})^x = x^{(3/2) \cdot x} \]
Thus, the equation becomes:
\[ x^{x^{3/2}} = x^{(3/2) \cdot x} \]

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

### Step 3: Rearrange the equation
To eliminate the fraction, we can multiply both sides by 2:
\[ 2 \cdot x^{3/2} = 3x \]
This simplifies to:
\[ 2x^{3/2} - 3x = 0 \]

### Step 4: Factor the equation
We can factor out \( x \) from the equation:
\[ x(2x^{1/2} - 3) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor equal to zero gives us:
1. \( x = 0 \) (not a valid solution since \( x \) cannot be zero in this context)
2. \( 2x^{1/2} - 3 = 0 \)

Now, solve for \( x \):
\[ 2x^{1/2} = 3 \]
\[ x^{1/2} = \frac{3}{2} \]
Squaring both sides gives:
\[ x = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

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