If `5^(sqrtx)+12^(sqrtx)=13^(sqrtx)`, the value of x is

Options

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

Answer

To solve the equation \( 5^{\sqrt{x}} + 12^{\sqrt{x}} = 13^{\sqrt{x}} \), we can follow these steps:

### Step 1: Recognize the Pythagorean Triplet
The numbers 5, 12, and 13 form a Pythagorean triplet. This means that \( 5^2 + 12^2 = 13^2 \).

### Step 2: Set Up the Equation
We can rewrite the equation as:
\[ 5^{\sqrt{x}} + 12^{\sqrt{x}} = 13^{\sqrt{x}} \]

### Step 3: Compare with Pythagorean Identity
Since \( 5^2 + 12^2 = 13^2 \), we can say that:
\[ (5^{\sqrt{x}})^2 + (12^{\sqrt{x}})^2 = (13^{\sqrt{x}})^2 \]
This suggests that the exponents must be equal for the equality to hold.

### Step 4: Set the Exponent Equal to 2
Since the bases are equal, we can set:
\[ \sqrt{x} = 2 \]

### Step 5: Solve for x
Now, we square both sides to find \( x \):
\[ x = (2)^2 = 4 \]

### Step 6: Verify the Solution
We can check if \( x = 4 \) satisfies the original equation:
\[ 5^{\sqrt{4}} + 12^{\sqrt{4}} = 13^{\sqrt{4}} \]
Calculating gives:
\[ 5^2 + 12^2 = 13^2 \]
\[ 25 + 144 = 169 \]
Since both sides are equal, our solution is verified.

### Final Answer
The value of \( x \) is \( 4 \).

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