Subjects algebra

Complex Division 50303B

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1. **State the problem:** Simplify the complex fraction $$\frac{3 - 3i}{-i}$$ and write the answer in the form $a + bi$. 2. **Recall the formula and rules:** To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator or simplify by removing imaginary unit from denominator. 3. **Multiply numerator and denominator by $i$ to remove $-i$ from denominator:** $$\frac{3 - 3i}{-i} \times \frac{i}{i} = \frac{(3 - 3i) i}{-i \times i}$$ 4. **Calculate numerator:** $$ (3 - 3i) i = 3i - 3i^2 $$ Recall that $i^2 = -1$, so: $$ 3i - 3(-1) = 3i + 3 $$ 5. **Calculate denominator:** $$ -i \times i = -i^2 = -(-1) = 1 $$ 6. **Rewrite the fraction:** $$ \frac{3 + 3i}{1} = 3 + 3i $$ 7. **Final answer:** $$ 3 + 3i $$