Which of the following value of x satisifies the equation : `4^(x-1).9^(2x-6)=5184` ?

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the equation \( 4^{(x-1)} \cdot 9^{(2x-6)} = 5184 \), we will follow these steps:

### Step 1: Rewrite the bases
We know that \( 4 \) can be expressed as \( 2^2 \) and \( 9 \) can be expressed as \( 3^2 \). Therefore, we can rewrite the equation as:
\[ (2^2)^{(x-1)} \cdot (3^2)^{(2x-6)} = 5184 \]
This simplifies to:
\[ 2^{2(x-1)} \cdot 3^{2(2x-6)} = 5184 \]

### Step 2: Simplify the exponents
Now, we can simplify the exponents:
\[ 2^{2x-2} \cdot 3^{4x-12} = 5184 \]

### Step 3: Factor 5184
Next, we need to express \( 5184 \) in terms of its prime factors. We can find that:
\[ 5184 = 2^6 \cdot 3^4 \]

### Step 4: Set up the equation
Now we can equate the powers of the bases:
\[ 2^{2x-2} \cdot 3^{4x-12} = 2^6 \cdot 3^4 \]
From this, we can set up two separate equations:
1. \( 2x - 2 = 6 \)
2. \( 4x - 12 = 4 \)

### Step 5: Solve the first equation
Let's solve the first equation:
\[ 2x - 2 = 6 \]
Adding \( 2 \) to both sides:
\[ 2x = 8 \]
Dividing by \( 2 \):
\[ x = 4 \]

### Step 6: Solve the second equation
Now, let's solve the second equation:
\[ 4x - 12 = 4 \]
Adding \( 12 \) to both sides:
\[ 4x = 16 \]
Dividing by \( 4 \):
\[ x = 4 \]

### Conclusion
Both equations yield the same solution, \( x = 4 \). Therefore, the value of \( x \) that satisfies the equation \( 4^{(x-1)} \cdot 9^{(2x-6)} = 5184 \) is:
\[ \boxed{4} \]