If `16^(2x-6)=64`, then. What is the value of x.

Options

A. a. `-9//4`
B. b. `9//4`
C. c. `-18//4`
D. d. `15//4`

Answer

To solve the equation \( 16^{(2x-6)} = 64 \), we can follow these steps:

### Step 1: Rewrite the bases
First, we need to express both sides of the equation with the same base. We know that:
- \( 16 = 4^2 \)
- \( 64 = 4^3 \)

So we can rewrite the equation as:
\[ (4^2)^{(2x-6)} = 4^3 \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify the left side:
\[ 4^{2(2x-6)} = 4^3 \]

This simplifies to:
\[ 4^{(4x - 12)} = 4^3 \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 4x - 12 = 3 \]

### Step 4: Solve for \(x\)
Now, we will solve for \(x\):
1. Add 12 to both sides:
\[ 4x = 3 + 12 \]
\[ 4x = 15 \]

2. Divide both sides by 4:
\[ x = \frac{15}{4} \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{\frac{15}{4}} \]

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