Find the value of x if `[3^(2x-2) +10] // 13=7`

Options

A. 1
B. 3
C. 4
D. 2

Answer

To solve the equation \(\frac{3^{(2x-2)} + 10}{13} = 7\), we will follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by 13 to eliminate the denominator.

\[ 3^{(2x-2)} + 10 = 7 \times 13 \]

### Step 2: Calculate the right side
Calculate \(7 \times 13\).

\[ 3^{(2x-2)} + 10 = 91 \]

### Step 3: Isolate the exponential term
Subtract 10 from both sides to isolate the term with the exponent.

\[ 3^{(2x-2)} = 91 - 10 \]

### Step 4: Simplify the right side
Calculate \(91 - 10\).

\[ 3^{(2x-2)} = 81 \]

### Step 5: Express 81 as a power of 3
Recognize that \(81\) can be expressed as a power of \(3\).

\[ 81 = 3^4 \]

### Step 6: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other.

\[ 2x - 2 = 4 \]

### Step 7: Solve for \(x\)
Add \(2\) to both sides.

\[ 2x = 4 + 2 \]

### Step 8: Simplify
\[ 2x = 6 \]

### Step 9: Divide by 2
Divide both sides by \(2\) to solve for \(x\).

\[ x = \frac{6}{2} \]

### Step 10: Final answer
\[ x = 3 \]

Thus, the value of \(x\) is \(3\).

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