1. **State the problem:** Simplify the expression involving complex numbers and write the answer in standard form $a + bi$. 2. **Expression given:** $$\frac{3 + 7i \cdot (1 + 2i)}{1 - 2i \cdot (1 + 2i)} + \frac{(3 + i)(1 + 2i)}{(-2i)(1 + 2i)} + \frac{1}{i} + \frac{1}{(2ji)^3} \cdot 5$$ 3. **Step 1: Simplify each part separately.** - For the first fraction numerator: $$7i \cdot (1 + 2i) = 7i + 14i^2 = 7i - 14$$ (since $i^2 = -1$) - So numerator: $$3 + (7i - 14) = (3 - 14) + 7i = -11 + 7i$$ - For the first fraction denominator: $$1 - 2i \cdot (1 + 2i) = 1 - 2i - 4i^2 = 1 - 2i + 4 = 5 - 2i$$ - First fraction: $$\frac{-11 + 7i}{5 - 2i}$$ 4. **Rationalize the first fraction denominator:** Multiply numerator and denominator by conjugate of denominator $5 + 2i$: $$\frac{(-11 + 7i)(5 + 2i)}{(5 - 2i)(5 + 2i)}$$ Calculate numerator: $$-11 \cdot 5 = -55$$ $$-11 \cdot 2i = -22i$$ $$7i \cdot 5 = 35i$$ $$7i \cdot 2i = 14i^2 = -14$$ Sum numerator: $$-55 - 22i + 35i - 14 = (-55 - 14) + (13i) = -69 + 13i$$ Denominator: $$5^2 - (2i)^2 = 25 - (-4) = 25 + 4 = 29$$ So first fraction simplified: $$\frac{-69 + 13i}{29} = -\frac{69}{29} + \frac{13}{29}i$$ 5. **Simplify second fraction:** $$(3 + i)(1 + 2i) = 3 + 6i + i + 2i^2 = 3 + 7i - 2 = 1 + 7i$$ Denominator: $$(-2i)(1 + 2i) = -2i - 4i^2 = -2i + 4 = 4 - 2i$$ Rationalize denominator by multiplying numerator and denominator by conjugate $4 + 2i$: Numerator: $$(1 + 7i)(4 + 2i) = 4 + 2i + 28i + 14i^2 = 4 + 30i - 14 = -10 + 30i$$ Denominator: $$(4 - 2i)(4 + 2i) = 16 - (2i)^2 = 16 - (-4) = 20$$ Second fraction simplified: $$\frac{-10 + 30i}{20} = -\frac{1}{2} + \frac{3}{2}i$$ 6. **Simplify $\frac{1}{i}$:** Multiply numerator and denominator by $i$: $$\frac{1}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$$ 7. **Simplify $\frac{1}{(2ji)^3} \cdot 5$:** Assuming $j = i$ (common in complex numbers), then: $$(2ji)^3 = (2i \cdot i)^3 = (2i^2)^3 = (2 \cdot -1)^3 = (-2)^3 = -8$$ So: $$\frac{1}{-8} \cdot 5 = -\frac{5}{8}$$ 8. **Sum all parts:** $$\left(-\frac{69}{29} + \frac{13}{29}i\right) + \left(-\frac{1}{2} + \frac{3}{2}i\right) + (-i) + \left(-\frac{5}{8}\right)$$ Combine real parts: $$-\frac{69}{29} - \frac{1}{2} - \frac{5}{8}$$ Find common denominator 232 (LCM of 29, 2, 8): $$-\frac{69}{29} = -\frac{69 \times 8}{232} = -\frac{552}{232}$$ $$-\frac{1}{2} = -\frac{116}{232}$$ $$-\frac{5}{8} = -\frac{145}{232}$$ Sum real: $$-\frac{552 + 116 + 145}{232} = -\frac{813}{232}$$ Combine imaginary parts: $$\frac{13}{29}i + \frac{3}{2}i - i = \left(\frac{13}{29} + \frac{3}{2} - 1\right)i$$ Convert to common denominator 58: $$\frac{13}{29} = \frac{26}{58}$$ $$\frac{3}{2} = \frac{87}{58}$$ $$1 = \frac{58}{58}$$ Sum imaginary: $$\left(\frac{26}{58} + \frac{87}{58} - \frac{58}{58}\right)i = \frac{55}{58}i$$ 9. **Final answer in standard form:** $$-\frac{813}{232} + \frac{55}{58}i$$ --- **Slug:** complex expression **Subject:** algebra **Desmos:** {"latex":"y=0","features":{"intercepts":true,"extrema":true}} **q_count:** 1