Write each expression in the form \( e^{k x} \) for a suitable con- stant \( k \). \( \begin{array}{ll}\text { 1.3. }\left(e^{2}\right)^{x},\left(\frac{1}{e}\right)^{x} & \text { 14. }\left(e^{3}\right)^{x / 5},\left(\frac{1}{e^{2}}\right)^{x} \\ \text { 15. }\left(\frac{1}{e^{3}}\right)^{2 x}, e^{1-x} \cdot e^{3 x-1} & \text { 16. }\left(\frac{e^{5}}{e^{3}}\right)^{x}, e^{4 x+2} \cdot e^{x-2}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{1}{e}\right)^{x}\) - step1: Expand the expression: \(\frac{1}{e^{x}}\) Calculate or simplify the expression \( (e^2)^x \). Simplify the expression by following steps: - step0: Solution: \(\left(e^{2}\right)^{x}\) - step1: Multiply the exponents: \(e^{2x}\) Calculate or simplify the expression \( e^(4*x+2) * e^(x-2) \). Simplify the expression by following steps: - step0: Solution: \(e^{4x+2}\times e^{x-2}\) - step1: Multiply: \(e^{4x+2+x-2}\) - step2: Calculate: \(e^{5x}\) Calculate or simplify the expression \( (1/e^3)^(2*x) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{1}{e^{3}}\right)^{2x}\) - step1: Expand the expression: \(\frac{1}{e^{6x}}\) Calculate or simplify the expression \( (e^5/e^3)^x \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{e^{5}}{e^{3}}\right)^{x}\) - step1: Multiply by \(a^{-n}:\) \(\left(e^{5}\times e^{-3}\right)^{x}\) - step2: Multiply the numbers: \(\left(e^{2}\right)^{x}\) - step3: Transform the expression: \(e^{2x}\) Calculate or simplify the expression \( e^(1-x) * e^(3*x-1) \). Simplify the expression by following steps: - step0: Solution: \(e^{1-x}\times e^{3x-1}\) - step1: Multiply: \(e^{1-x+3x-1}\) - step2: Calculate: \(e^{2x}\) Calculate or simplify the expression \( (e^3)^(x/5) \). Simplify the expression by following steps: - step0: Solution: \(\left(e^{3}\right)^{\frac{x}{5}}\) - step1: Multiply the exponents: \(e^{3\times \frac{x}{5}}\) - step2: Multiply the terms: \(e^{\frac{3x}{5}}\) Calculate or simplify the expression \( (1/e^2)^x \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{1}{e^{2}}\right)^{x}\) - step1: Expand the expression: \(\frac{1}{e^{2x}}\) Here are the expressions rewritten in the form \( e^{k x} \) for suitable constants \( k \): 1. \( (e^2)^x = e^{2x} \) 2. \( \left(\frac{1}{e}\right)^x = \frac{1}{e^x} = e^{-x} \) 3. \( (e^3)^{x/5} = e^{\frac{3x}{5}} \) 4. \( \left(\frac{1}{e^2}\right)^x = \frac{1}{e^{2x}} = e^{-2x} \) 5. \( \left(\frac{1}{e^3}\right)^{2x} = \frac{1}{e^{6x}} = e^{-6x} \) 6. \( e^{1-x} \cdot e^{3x-1} = e^{(1-x) + (3x-1)} = e^{2x} \) 7. \( \left(\frac{e^5}{e^3}\right)^x = e^{(5-3)x} = e^{2x} \) 8. \( e^{4x+2} \cdot e^{x-2} = e^{(4x+2) + (x-2)} = e^{5x} \) Thus, the final forms are: - \( e^{2x} \) - \( e^{-x} \) - \( e^{\frac{3x}{5}} \) - \( e^{-2x} \) - \( e^{-6x} \) - \( e^{2x} \) - \( e^{2x} \) - \( e^{5x} \)