d) \( \frac{4^{x} \cdot 2^{x+2}-8^{x}}{2^{2} \cdot 3^{0} \cdot 2^{3 x}} \)

Let's simplify the expression step by step while keeping things fun! First off, we know that \( 4^x \) can be rewritten as \( (2^2)^x = 2^{2x} \), and \( 8^x \) can be rewritten as \( (2^3)^x = 2^{3x} \). Thus, the numerator becomes: \[ 4^x \cdot 2^{x+2} - 8^x = 2^{2x} \cdot 2^{x+2} - 2^{3x} = 2^{2x + x + 2} - 2^{3x} = 2^{3x + 2} - 2^{3x} \] Now, we can factor out \( 2^{3x} \) from the numerator: \[ 2^{3x}(2^2 - 1) = 2^{3x}(4 - 1) = 2^{3x} \cdot 3 \] Next, let's simplify the denominator: \[ 2^2 \cdot 3^0 \cdot 2^{3x} = 4 \cdot 1 \cdot 2^{3x} = 4 \cdot 2^{3x} \] Now, putting it all together: \[ \frac{2^{3x} \cdot 3}{4 \cdot 2^{3x}} \] When you simplify this, the \( 2^{3x} \) terms cancel out: \[ \frac{3}{4} \] So the simplified expression is: \[ \frac{3}{4} \] And there you have it! Simple yet effective! 😊