Simplify each expression as far as possible: (a) \(\mathrm{e}^{2 x} \mathrm{e}^{7 x}\) (b) \(\left(3 \mathrm{e}^{x}\right)\left(2 \mathrm{e}^{-x}\right)\) (c) \(\mathrm{e}^{2 x}\left(\mathrm{e}^{-2 x}+\mathrm{e}^{-x}+1\right)-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) (d) \(\frac{\mathrm{e}^{-3 x}}{2 \mathrm{e}^{-x}}\)

Given the expression \(\mathrm{e}^{2 x} \mathrm{e}^{7 x}\), apply the properties of exponents: \(\mathrm{e}^{2x} \cdot \mathrm{e}^{7x} = \mathrm{e}^{2x + 7x} = \boxed{\mathrm{e}^{9x}}\).

For the expression \(\left(3 \mathrm{e}^{x}\right)\left(2 \mathrm{e}^{-x}\right)\), apply the properties of exponents and multiplication: \((3 \mathrm{e}^{x})(2 \mathrm{e}^{-x}) = 3 \cdot 2 \cdot \mathrm{e}^{x} \cdot \mathrm{e}^{-x} = 6 \cdot \mathrm{e}^{x-x} = \boxed{6}\).

Simplify the expression \(\mathrm{e}^{2x}\left(\mathrm{e}^{-2x}+\mathrm{e}^{-x}+1\right)-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) as follows: 1. Distribute \(\mathrm{e}^{2x}\) across the terms within the parentheses: \(\mathrm{e}^{2x}\mathrm{e}^{-2x}+\mathrm{e}^{2x}\mathrm{e}^{-x}+\mathrm{e}^{2x}-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) 2. Apply the properties of exponents: \(1+\mathrm{e}^{x}+\mathrm{e}^{2x}-\mathrm{e}^{x}\left(1+\mathrm{e}^{x}\right)\) 3. Distribute \(-\mathrm{e}^{x}\) across the terms within the parentheses: \(1+\mathrm{e}^{x}+\mathrm{e}^{2x}-\mathrm{e}^{x}-\mathrm{e}^{2x}\) 4. Combine like terms: \(1 + \left(\mathrm{e}^{x} - \mathrm{e}^{x}\right) + \left(\mathrm{e}^{2x} - \mathrm{e}^{2x}\right) = \boxed{1}\).

Finally, simplify the expression \(\frac{\mathrm{e}^{-3x}}{2\mathrm{e}^{-x}}\): 1. Apply the properties of exponents: \(\frac{\mathrm{e}^{-3x}}{2\mathrm{e}^{-x}} = \frac{1}{2} \cdot \frac{\mathrm{e}^{-3x}}{\mathrm{e}^{-x}}\). 2. Continue applying the properties of exponents: \(\frac{1}{2} \cdot \mathrm{e}^{-3x + x} = \boxed{\frac{1}{2}\mathrm{e}^{-2x}}\).