Solve for x : `x:(13)^sqrt(x)=4^(4)-3^(4)-6.`

Answer

To solve the equation \( x:(13)^{\sqrt{x}}=4^{4}-3^{4}-6 \), we will follow these steps:

### Step 1: Write the equation
We start with the equation:
\[ (13)^{\sqrt{x}} = 4^{4} - 3^{4} - 6 \]

### Step 2: Simplify the right-hand side
First, we need to calculate \( 4^{4} \) and \( 3^{4} \):
\[ 4^{4} = 256 \quad \text{and} \quad 3^{4} = 81 \]
Now substitute these values back into the equation:
\[ (13)^{\sqrt{x}} = 256 - 81 - 6 \]

### Step 3: Perform the subtraction
Now, we simplify the right-hand side:
\[ 256 - 81 = 175 \]
Then,
\[ 175 - 6 = 169 \]
So, we have:
\[ (13)^{\sqrt{x}} = 169 \]

### Step 4: Express 169 as a power of 13
Next, we can express 169 as a power of 13:
\[ 169 = 13^{2} \]
Thus, we can rewrite the equation as:
\[ (13)^{\sqrt{x}} = (13)^{2} \]

### Step 5: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ \sqrt{x} = 2 \]

### Step 6: Solve for \( x \)
Now, we square both sides to solve for \( x \):
\[ x = 2^{2} = 4 \]

### Final Answer
Thus, the solution for \( x \) is:
\[ \boxed{4} \]