Find the value of x, if `(2^(x-1)*4^(2x+1))/(8^(x-1))=64`.

Options

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

Answer

To solve the equation \((2^{x-1} \cdot 4^{2x+1})/(8^{x-1}) = 64\), we will convert all terms to base 2 and then simplify.

### Step-by-Step Solution:

1. **Convert all numbers to base 2**:
- We know that \(4 = 2^2\) and \(8 = 2^3\).
- Therefore, we can rewrite the equation:
\[ 4^{2x+1} = (2^2)^{2x+1} = 2^{2(2x+1)} = 2^{4x + 2} \]
\[ 8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} = 2^{3x - 3} \]
- The right side, \(64\), can be written as:
\[ 64 = 2^6 \]

2. **Substituting back into the equation**:
- Now substituting these back into the original equation gives us:
\[ \frac{2^{x-1} \cdot 2^{4x + 2}}{2^{3x - 3}} = 2^6 \]

3. **Combine the exponents in the numerator**:
- The numerator can be combined using the property of exponents:
\[ 2^{(x-1) + (4x + 2)} = 2^{5x + 1} \]
- So, the equation now looks like:
\[ \frac{2^{5x + 1}}{2^{3x - 3}} = 2^6 \]

4. **Simplifying the left side**:
- Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[ 2^{(5x + 1) - (3x - 3)} = 2^6 \]
- Simplifying the exponent gives:
\[ 2^{5x + 1 - 3x + 3} = 2^6 \]
\[ 2^{2x + 4} = 2^6 \]

5. **Setting the exponents equal**:
- Since the bases are the same, we can set the exponents equal to each other:
\[ 2x + 4 = 6 \]

6. **Solving for \(x\)**:
- Subtract 4 from both sides:
\[ 2x = 2 \]
- Divide by 2:
\[ x = 1 \]

### Final Answer:
The value of \(x\) is \(1\).