Vermenigvuldigen en delen (a+bi)

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Bereken

1. \(\frac{9-i}{-9+9i}\)
2. \((4-10i)\cdot (+3i)\)
3. \((+i) \cdot (5+7i)\)
4. \((+9i) \cdot (1-3i)\)
5. \((6-3i) \cdot (9+5i)\)
6. \((-10+i) \cdot (10+9i)\)
7. \(\frac{6+8i}{9-7i}\)
8. \(\frac{-9-3i}{-3-5i}\)
9. \(\frac{-1+7i}{-10+6i}\)
10. \((3+i) \cdot (10+10i)\)
11. \(\frac{10+8i}{6+3i}\)
12. \(\frac{6+5i}{7+6i}\)

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Bereken

Verbetersleutel

1. \(\frac{9-i}{-9+9i}= \frac{9-i}{-9+9i} \cdot \frac{-9-9i}{-9-9i} = \frac{-81-81i +9 i+9i^2 }{(-9)^2-(9i)^2} = \frac{-81-81i +9 i-9}{81 + 81} = \frac{-90-72i }{162} = \frac{-5}{9} + \frac{-4}{9}i \)
2. \((4-10i)\cdot (+3i)= +12 i-30i^2 = \color{red}{30}\color{blue}{+12i}\)
3. \((+i) \cdot (5+7i)= +5 i+7i^2 = \color{red}{-7}\color{blue}{+5i}\)
4. \((+9i) \cdot (1-3i)= +9 i-27i^2 = \color{red}{27}\color{blue}{+9i}\)
5. \((6-3i) \cdot (9+5i)= 54+30i -27 i-15i^2 = 54+30i -27 i+15= \color{red}{54+15}\color{blue}{+30i -27i}=\color{red}{69}\color{blue}{+3i}\)
6. \((-10+i) \cdot (10+9i)= -100-90i +10 i+9i^2 = -100-90i +10 i-9= \color{red}{-100-9}\color{blue}{-90i +10i}=\color{red}{-109}\color{blue}{-80i}\)
7. \(\frac{6+8i}{9-7i}= \frac{6+8i}{9-7i} \cdot \frac{9+7i}{9+7i} = \frac{54+42i +72 i+56i^2 }{(9)^2-(-7i)^2} = \frac{54+42i +72 i-56}{81 + 49} = \frac{-2+114i }{130} = \frac{-1}{65} - \frac{-57}{65}i \)
8. \(\frac{-9-3i}{-3-5i}= \frac{-9-3i}{-3-5i} \cdot \frac{-3+5i}{-3+5i} = \frac{27-45i +9 i-15i^2 }{(-3)^2-(-5i)^2} = \frac{27-45i +9 i+15}{9 + 25} = \frac{42-36i }{34} = \frac{21}{17} + \frac{-18}{17}i \)
9. \(\frac{-1+7i}{-10+6i}= \frac{-1+7i}{-10+6i} \cdot \frac{-10-6i}{-10-6i} = \frac{10+6i -70 i-42i^2 }{(-10)^2-(6i)^2} = \frac{10+6i -70 i+42}{100 + 36} = \frac{52-64i }{136} = \frac{13}{34} + \frac{-8}{17}i \)
10. \((3+i) \cdot (10+10i)= 30+30i +10 i+10i^2 = 30+30i +10 i-10= \color{red}{30-10}\color{blue}{+30i +10i}=\color{red}{20}\color{blue}{+40i}\)
11. \(\frac{10+8i}{6+3i}= \frac{10+8i}{6+3i} \cdot \frac{6-3i}{6-3i} = \frac{60-30i +48 i-24i^2 }{(6)^2-(3i)^2} = \frac{60-30i +48 i+24}{36 + 9} = \frac{84+18i }{45} = \frac{28}{15} - \frac{-2}{5}i \)
12. \(\frac{6+5i}{7+6i}= \frac{6+5i}{7+6i} \cdot \frac{7-6i}{7-6i} = \frac{42-36i +35 i-30i^2 }{(7)^2-(6i)^2} = \frac{42-36i +35 i+30}{49 + 36} = \frac{72-i }{85} = \frac{72}{85} + \frac{-1}{85}i \)