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 follow these steps:

### Step 1: Take logarithm of 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 the logarithmic identity
Using the property of logarithms, \( \log_a (b^m) = m \log_a (b) \), we can simplify both sides:
\[ x \sqrt{x} \cdot \log_x (x) = x \cdot \log_x (x \sqrt{x}) \]

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

### Step 4: Set the equation
Now we have:
\[ x \sqrt{x} = \frac{3x}{2} \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ x \sqrt{x} - \frac{3x}{2} = 0 \]
Factoring out \( x \):
\[ x \left( \sqrt{x} - \frac{3}{2} \right) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x = 0 \)
2. \( \sqrt{x} = \frac{3}{2} \) which implies \( x = \left( \frac{3}{2} \right)^2 = \frac{9}{4} \)

### Step 7: Check for valid solutions
Since we took logarithms, \( x \) must be greater than 0. Therefore, we discard \( x = 0 \).

### Step 8: Verify if \( x = 1 \) is a solution
We can also check if \( x = 1 \) satisfies the original equation:
\[ 1^{1 \cdot \sqrt{1}} = (1 \cdot \sqrt{1})^1 \implies 1 = 1 \]
Thus, \( x = 1 \) is also a valid solution.

### Conclusion
The valid solutions we have are:
- \( x = 1 \)
- \( x = \frac{9}{4} \)

Thus, the number of solutions to the equation is **2**.

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