Number of solution(s) of the equation `x^(x sqrt(x))=(x sqrt(x))^(x)` is/are :

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( x^{(x \sqrt{x})} = (x \sqrt{x})^{x} \), we will take the logarithm of both sides and simplify the equation step by step.

### Step 1: Take logarithm on both sides
We start by taking the logarithm of both sides with base \( x \):
\[ \log_x (x^{(x \sqrt{x})}) = \log_x ((x \sqrt{x})^{x}) \]

### Step 2: Apply logarithmic properties
Using the property \( \log_a (b^c) = c \cdot \log_a (b) \), we can simplify both sides:
\[ x \sqrt{x} = x \cdot \log_x (x \sqrt{x}) \]

### Step 3: Simplify the right side
Now, we simplify \( \log_x (x \sqrt{x}) \):
\[ \log_x (x \sqrt{x}) = \log_x (x) + \log_x (\sqrt{x}) = 1 + \frac{1}{2} = \frac{3}{2} \]
Thus, we have:
\[ x \sqrt{x} = x \cdot \frac{3}{2} \]

### Step 4: Eliminate \( x \) from both sides
Assuming \( x \neq 0 \), we can divide both sides by \( x \):
\[ \sqrt{x} = \frac{3}{2} \]

### Step 5: Solve for \( x \)
Now, squaring both sides gives:
\[ x = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 6: Check for other solutions
We also need to check if \( x = 1 \) is a solution:
\[ 1^{(1 \cdot \sqrt{1})} = (1 \cdot \sqrt{1})^{1} \implies 1 = 1 \]
So, \( x = 1 \) is indeed a solution.

### Step 7: Consider the domain of \( x \)
Since \( x \) is in the base of the logarithm, \( x \) must be greater than 0 and not equal to 1. Therefore, \( x = 0 \) is not valid, and we discard it.

### Final Solutions
The valid solutions are:
1. \( x = 1 \)
2. \( x = \frac{9}{4} \)

Thus, the total number of solutions is **2**.