1. **State the problem:** We want to simplify the complex expression $$A = e^{i\phi} \left(e^{i\theta_1} - 1\right) + e^{i(\theta_1 + \theta_2 + \theta_3)} - e^{i(\theta_1 + \theta_2)}$$ 2. **Recall important rules:** - Exponentials with imaginary exponents represent points on the unit circle in the complex plane. - We can factor and combine terms using properties of exponents: $$e^{a} e^{b} = e^{a+b}$$ - The goal is to factor and simplify the expression as much as possible. 3. **Rewrite the expression grouping terms:** $$A = e^{i\phi} e^{i\theta_1} - e^{i\phi} + e^{i(\theta_1 + \theta_2 + \theta_3)} - e^{i(\theta_1 + \theta_2)}$$ 4. **Factor where possible:** - From the last two terms: $$e^{i(\theta_1 + \theta_2 + \theta_3)} - e^{i(\theta_1 + \theta_2)} = e^{i(\theta_1 + \theta_2)} \left(e^{i\theta_3} - 1\right)$$ - From the first two terms: $$e^{i\phi} e^{i\theta_1} - e^{i\phi} = e^{i\phi} \left(e^{i\theta_1} - 1\right)$$ 5. **Rewrite $A$ using these factors:** $$A = e^{i\phi} \left(e^{i\theta_1} - 1\right) + e^{i(\theta_1 + \theta_2)} \left(e^{i\theta_3} - 1\right)$$ 6. **Interpretation:** The expression is now a sum of two terms, each a product of a complex exponential and a difference of complex exponentials. This form is often useful in physics and engineering for analyzing phase differences. **Final simplified form:** $$\boxed{A = e^{i\phi} \left(e^{i\theta_1} - 1\right) + e^{i(\theta_1 + \theta_2)} \left(e^{i\theta_3} - 1\right)}$$