If `3^(3sqrt(x)) + 4^(3sqrt(x)) = 5^(3sqrt(x))` then the value of x is :

Options

A. 8
B. 2
C. 4
D. 1

Answer

To solve the equation \( 3^{3\sqrt{x}} + 4^{3\sqrt{x}} = 5^{3\sqrt{x}} \), we can proceed with the following steps:

### Step 1: Let \( y = 3\sqrt{x} \)

We start by substituting \( y \) for \( 3\sqrt{x} \). This gives us the equation:

\[ 3^y + 4^y = 5^y \]

### Step 2: Analyze the equation

We can recognize that this equation resembles the Pythagorean theorem, where \( 3^2 + 4^2 = 5^2 \). This suggests that \( y \) might have a specific value that satisfies this equation.

### Step 3: Test \( y = 2 \)

Let’s test if \( y = 2 \) satisfies the equation:

\[ 3^2 + 4^2 = 5^2 \]

Calculating each term:

\[ 9 + 16 = 25 \]

This is true, so \( y = 2 \) is indeed a solution.

### Step 4: Substitute back to find \( x \)

Recall that we set \( y = 3\sqrt{x} \). Now we can substitute back to find \( x \):

\[ 3\sqrt{x} = 2 \]

### Step 5: Solve for \( \sqrt{x} \)

Now, divide both sides by 3:

\[ \sqrt{x} = \frac{2}{3} \]

### Step 6: Square both sides to find \( x \)

Finally, square both sides to solve for \( x \):

\[ x = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

### Final Answer

Thus, the value of \( x \) is:

\[ \boxed{\frac{4}{9}} \]

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