1. **State the problem:** We need to find the length of the segment AB where A is the x-intercept and B is the y-intercept of the tangent line to the curve $y = e^x$ at the point $(2, e^2)$. 2. **Find the slope of the tangent line:** The derivative of $y = e^x$ is $\frac{dy}{dx} = e^x$. At $x=2$, the slope is $m = e^2$. 3. **Write the equation of the tangent line:** Using point-slope form: $$y - e^2 = e^2(x - 2)$$ Simplify: $$y = e^2 x - 2 e^2 + e^2 = e^2 x - e^2$$ 4. **Find the x-intercept (point A):** Set $y=0$: $$0 = e^2 x - e^2$$ $$e^2 x = e^2$$ $$x = 1$$ So, $A = (1, 0)$. 5. **Find the y-intercept (point B):** Set $x=0$: $$y = e^2 (0) - e^2 = -e^2$$ So, $B = (0, -e^2)$. 6. **Calculate the length AB:** Use the distance formula: $$AB = \sqrt{(1 - 0)^2 + (0 - (-e^2))^2} = \sqrt{1^2 + (e^2)^2} = \sqrt{1 + e^{4}}$$ **Final answer:** $$\boxed{\sqrt{1 + e^{4}}}$$