1. **Stating the problem:** We are given complex numbers $z_1 = -53i$ and $z_2 = 4 - 3i$. We need to find: a) $3 \cdot z_1 + 8 \cdot z_2$ b) $3 \cdot z_1 \cdot z_2$ c) $\frac{z_1}{z_2}$ 2. **Recall the form and rules:** Complex numbers are in the form $a + bi$ where $a,b \in \mathbb{R}$. Multiplication and addition follow algebraic rules with $i^2 = -1$. Division uses the conjugate of the denominator. --- ### a) Calculate $3 \cdot z_1 + 8 \cdot z_2$ $$3 \cdot z_1 = 3 \cdot (-53i) = -159i$$ $$8 \cdot z_2 = 8 \cdot (4 - 3i) = 32 - 24i$$ Add them: $$-159i + 32 - 24i = 32 + (-159 - 24)i = 32 - 183i$$ Answer: $32 - 183i$ --- ### b) Calculate $3 \cdot z_1 \cdot z_2$ First find $z_1 \cdot z_2$: $$z_1 \cdot z_2 = (-53i)(4 - 3i) = -53i \cdot 4 + (-53i)(-3i) = -212i + 159i^2$$ Recall $i^2 = -1$, so: $$159i^2 = 159 \times (-1) = -159$$ Thus: $$z_1 \cdot z_2 = -212i - 159 = -159 - 212i$$ Now multiply by 3: $$3 \cdot z_1 \cdot z_2 = 3(-159 - 212i) = -477 - 636i$$ Answer: $-477 - 636i$ --- ### c) Calculate $\frac{z_1}{z_2}$ Given: $$z_1 = -53i, \quad z_2 = 4 - 3i$$ To divide, multiply numerator and denominator by the conjugate of denominator: $$\frac{z_1}{z_2} = \frac{-53i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i} = \frac{-53i(4 + 3i)}{(4 - 3i)(4 + 3i)}$$ Calculate numerator: $$-53i \cdot 4 = -212i$$ $$-53i \cdot 3i = -159i^2 = -159 \times (-1) = 159$$ So numerator: $$-212i + 159 = 159 - 212i$$ Calculate denominator: $$(4)^2 - (3i)^2 = 16 - 9i^2 = 16 - 9(-1) = 16 + 9 = 25$$ Divide numerator by denominator: $$\frac{159 - 212i}{25} = \frac{159}{25} - \frac{212}{25}i$$ Answer: $\frac{159}{25} - \frac{212}{25}i$ --- **Final answers:** a) $32 - 183i$ b) $-477 - 636i$ c) $\frac{159}{25} - \frac{212}{25}i$