1.1 Simplify the following: \( \frac{2^{2013+2 x}-6.2^{2011+2 x}}{4^{1010+x}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2^{2013+2x}-6\times 2^{2011+2x}\right)}{4^{1010+x}}\) - step1: Remove the parentheses: \(\frac{2^{2013+2x}-6\times 2^{2011+2x}}{4^{1010+x}}\) - step2: Subtract the terms: \(\frac{-2^{2x+2012}}{4^{1010+x}}\) - step3: Calculate: \(\frac{-2^{2x+2012}}{\left(2^{2}\right)^{1010+x}}\) - step4: Calculate: \(\frac{-2^{2x+2012}}{2^{2020+2x}}\) - step5: Calculate: \(-2^{-8}\) To simplify the expression \[ \frac{2^{2013 + 2x} - 6 \cdot 2^{2011 + 2x}}{4^{1010 + x}}, \] we can follow these steps: 1. **Rewrite the denominator**: Notice that \(4^{1010 + x}\) can be rewritten in terms of base 2: \[ 4^{1010 + x} = (2^2)^{1010 + x} = 2^{2(1010 + x)} = 2^{2020 + 2x}. \] 2. **Rewrite the numerator**: The numerator can be factored: \[ 2^{2013 + 2x} - 6 \cdot 2^{2011 + 2x} = 2^{2011 + 2x}(2^2 - 6) = 2^{2011 + 2x}(4 - 6) = 2^{2011 + 2x}(-2). \] 3. **Combine the expressions**: Now substitute the rewritten numerator and denominator back into the expression: \[ \frac{2^{2011 + 2x}(-2)}{2^{2020 + 2x}}. \] 4. **Simplify the fraction**: When dividing powers of the same base, we subtract the exponents: \[ = -2 \cdot \frac{2^{2011 + 2x}}{2^{2020 + 2x}} = -2 \cdot 2^{(2011 + 2x) - (2020 + 2x)} = -2 \cdot 2^{-9} = -2^{-8}. \] Thus, the simplified form of the expression is \[ -2^{-8}. \]