Vermenigvuldigen en delen (a+bi)

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Bereken

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

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Bereken

Verbetersleutel

1. \((5-3i) \cdot (8-3i)= 40-15i -24 i+9i^2 = 40-15i -24 i-9= \color{red}{40-9}\color{blue}{-15i -24i}=\color{red}{31}\color{blue}{-39i}\)
2. \(\frac{7+7i}{-1+10i}= \frac{7+7i}{-1+10i} \cdot \frac{-1-10i}{-1-10i} = \frac{-7-70i -7 i-70i^2 }{(-1)^2-(10i)^2} = \frac{-7-70i -7 i+70}{1 + 100} = \frac{63-77i }{101} = \frac{63}{101} + \frac{-77}{101}i \)
3. \(\frac{7-8i}{-5+4i}= \frac{7-8i}{-5+4i} \cdot \frac{-5-4i}{-5-4i} = \frac{-35-28i +40 i+32i^2 }{(-5)^2-(4i)^2} = \frac{-35-28i +40 i-32}{25 + 16} = \frac{-67+12i }{41} = \frac{-67}{41} - \frac{-12}{41}i \)
4. \((-2-8i)\cdot (-2i)= +4 i+16i^2 = \color{red}{-16}\color{blue}{+4i}\)
5. \(\frac{-1-6i}{1+6i}= \frac{-1-6i}{1+6i} \cdot \frac{1-6i}{1-6i} = \frac{-1+6i -6 i+36i^2 }{(1)^2-(6i)^2} = \frac{-1+6i -6 i-36}{1 + 36} = \frac{-37+0i }{37} = -1+ 0i\)
6. \((+5i) \cdot (7-4i)= +35 i-20i^2 = \color{red}{20}\color{blue}{+35i}\)
7. \((10+3i)\cdot (+5i)= +50 i+15i^2 = \color{red}{-15}\color{blue}{+50i}\)
8. \(\frac{-2-9i}{-2-9i}= \frac{-2-9i}{-2-9i} \cdot \frac{-2+9i}{-2+9i} = \frac{4-18i +18 i-81i^2 }{(-2)^2-(-9i)^2} = \frac{4-18i +18 i+81}{4 + 81} = \frac{85+0i }{85} = 1+ 0i\)
9. \(\frac{1+10i}{5+7i}= \frac{1+10i}{5+7i} \cdot \frac{5-7i}{5-7i} = \frac{5-7i +50 i-70i^2 }{(5)^2-(7i)^2} = \frac{5-7i +50 i+70}{25 + 49} = \frac{75+43i }{74} = \frac{75}{74} - \frac{-43}{74}i \)
10. \((-6i) \cdot (-4-3i)= +24 i+18i^2 = \color{red}{-18}\color{blue}{+24i}\)
11. \(\frac{-4+3i}{7+7i}= \frac{-4+3i}{7+7i} \cdot \frac{7-7i}{7-7i} = \frac{-28+28i +21 i-21i^2 }{(7)^2-(7i)^2} = \frac{-28+28i +21 i+21}{49 + 49} = \frac{-7+49i }{98} = \frac{-1}{14} - \frac{-1}{2}i \)
12. \((-4+6i) \cdot (9+7i)= -36-28i +54 i+42i^2 = -36-28i +54 i-42= \color{red}{-36-42}\color{blue}{-28i +54i}=\color{red}{-78}\color{blue}{+26i}\)