Find the value of `x` if `4^(2x -3) = 4^(2) xx 2^(3) xx 4`.

Options

A. `0`
B. `4`
C. `(15)/(4)`
D. `(-9)/(8)`

Answer

To solve the equation \( 4^{2x - 3} = 4^2 \times 2^3 \times 4 \), we can follow these steps:

### Step 1: Rewrite the equation using powers of 2
We know that \( 4 = 2^2 \). Therefore, we can rewrite the left side and the right side of the equation in terms of base 2.

\[ 4^{2x - 3} = (2^2)^{2x - 3} \]

Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we have:

\[ (2^2)^{2x - 3} = 2^{2(2x - 3)} = 2^{4x - 6} \]

### Step 2: Simplify the right side
Now, let's simplify the right side of the equation:

\[ 4^2 \times 2^3 \times 4 = 4^2 \times 4 \times 2^3 \]

Again, rewriting \(4\) as \(2^2\):

\[ = (2^2)^2 \times 2^3 \times 2^2 \]

Using the power of a power property again:

\[ = 2^{2 \cdot 2} \times 2^3 \times 2^2 = 2^4 \times 2^3 \times 2^2 \]

### Step 3: Combine the powers on the right side
Now, we can combine the powers on the right side:

\[ = 2^{4 + 3 + 2} = 2^9 \]

### Step 4: Set the exponents equal to each other
Now we have:

\[ 2^{4x - 6} = 2^9 \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 4x - 6 = 9 \]

### Step 5: Solve for \(x\)
Now, we solve for \(x\):

\[ 4x = 9 + 6 \]
\[ 4x = 15 \]
\[ x = \frac{15}{4} \]

### Final Answer
Thus, the value of \(x\) is:

\[ \boxed{\frac{15}{4}} \]

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