If ` 5 ^(sqrt(x)) + 12 ^(sqrt(x)) = 13 ^(sqrt(x))` , then x is equal to

Options

A. A)`(25)/(4)`
B. B)4
C. C)9
D. D)16

Answer

To solve the equation \( 5^{\sqrt{x}} + 12^{\sqrt{x}} = 13^{\sqrt{x}} \), we can recognize that this resembles the Pythagorean theorem, where \( 5^2 + 12^2 = 13^2 \).

### Step-by-Step Solution:

1. **Rewrite the equation**:
\[ 5^{\sqrt{x}} + 12^{\sqrt{x}} = 13^{\sqrt{x}} \]

2. **Identify the Pythagorean relationship**:
We know that \( 5^2 + 12^2 = 13^2 \). This suggests that we can set \( \sqrt{x} = k \) for some \( k \).

3. **Set up the equation using \( k \)**:
\[ 5^k + 12^k = 13^k \]

4. **Recognize that this holds true for \( k = 2 \)**:
If we substitute \( k = 2 \):
\[ 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \]
This confirms that \( k = 2 \) satisfies the equation.

5. **Relate \( k \) back to \( x \)**:
Since we set \( \sqrt{x} = k \), we have:
\[ \sqrt{x} = 2 \]

6. **Square both sides to solve for \( x \)**:
\[ x = 2^2 = 4 \]

### Final Answer:
Thus, the value of \( x \) is \( 4 \).