Subjects algebra

Complex Multiplication E7E2Ec

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1. **State the problem:** Simplify the expression $$(3 - )(9 - 2i)(1 + 3i)$$. Since the first term is incomplete, assuming it is $3$ (without any imaginary part), the expression becomes $$3 \times (9 - 2i) \times (1 + 3i)$$. 2. **Recall the formula:** To multiply complex numbers, use the distributive property and remember that $$i^2 = -1$$. 3. **First multiply** $$(9 - 2i)(1 + 3i)$$: $$\begin{aligned} (9 - 2i)(1 + 3i) &= 9 \times 1 + 9 \times 3i - 2i \times 1 - 2i \times 3i \\ &= 9 + 27i - 2i - 6i^2 \\ &= 9 + 25i - 6(-1) \\ &= 9 + 25i + 6 \\ &= 15 + 25i \end{aligned}$$ 4. **Now multiply** $$3 \times (15 + 25i)$$: $$3 \times 15 + 3 \times 25i = 45 + 75i$$ 5. **Final answer:** $$\boxed{45 + 75i}$$