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

To simplify the expression \( \frac{2^{2013+2 x}-6.2^{2011+2 x}}{4^{1010+x}} \), we can start by rewriting \( 4 \) in terms of \( 2 \): \[ 4^{1010+x} = (2^2)^{1010+x} = 2^{2(1010+x)} = 2^{2020 + 2x} \] Now, let's rewrite the numerator: \[ 2^{2013 + 2x} - 6 \cdot 2^{2011 + 2x} \] Factoring out the common term \( 2^{2011 + 2x} \): \[ = 2^{2011 + 2x} \left( 2^2 - 6 \right) = 2^{2011 + 2x} (4 - 6) = 2^{2011 + 2x} (-2) \] Now, substituting back into the original expression, we have: \[ \frac{2^{2011 + 2x} (-2)}{2^{2020 + 2x}} \] This can be simplified by dividing the powers of \( 2 \): \[ = -2 \cdot \frac{2^{2011 + 2x}}{2^{2020 + 2x}} = -2 \cdot 2^{(2011 + 2x) - (2020 + 2x)} = -2 \cdot 2^{2011 - 2020} = -2 \cdot 2^{-9} = -\frac{2}{512} = -\frac{1}{256} \] Thus, the simplified form of the given expression is: \[ \boxed{-\frac{1}{256}} \]