Solutions to Problem Set 24

Math 101-06

11-8-2017

[Complex numbers]

1. Simplify $\sqrt{-25}$ .

$$\sqrt{-25} = \sqrt{25} \sqrt{-1} = 5 i.\quad\halmos$$


2. Simplify $\sqrt{-48}$ .

$$\sqrt{-48} = \sqrt{48} \sqrt{-1} = \sqrt{16} \sqrt{3} \sqrt{-1} = 4 i \sqrt{3}.\quad\halmos$$


3. Simplify $(11 + 3 i) + (8 - 15
   i)$ .

$$(11 + 3 i) + (8 - 15 i) = (11 + 8) + (3 i - 15 i) = 19 - 12 i.\quad\halmos$$


4. Simplify $(6 + 17 i) - (2 + 7
   i)$ .

$$(6 + 17 i) - (2 + 7 i) = (6 - 2) + (17 i - 7 i) = 4 + 10 i.\quad\halmos$$


5. Multiply out and simplify: $5
   i (6 - 2 i)$ .

$$5 i (6 - 2 i) = 5 i \cdot 6 - 5 i \cdot 2 i = 30 i - 10 i^2 = 30 i - 10 \cdot (-1) = 10 + 30 i.\quad\halmos$$


6. Simplify $i^{67}$ .

$$i^{67} = i^{66} \cdot i^1 = (i^2)^{33} \cdot i = (-1)^{33} \cdot i = -1 \cdot i = -i.\quad\halmos$$


7. Simplify $i^{82}$ .

$$i^{82} = (i^2)^{41} = (-1)^{41} = -1.\quad\halmos$$


8. Multiply out and simplify: $(2 - 5 i)(3 + i)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & 2 & & $-5 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 3 & & 6 & & $-15 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & i & & $2 i$ & & $-5 i^2 = 5$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(2 - 5 i)(3 + i) = 11 - 13 i.\quad\halmos$$


9. Multiply out and simplify: $(6 - 4 i)(6 + 4 i)$ .

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & 6 & & $-4 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 6 & & 36 & & $-24 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $4 i$ & & $24 i$ & & $-16 i^2 = 16$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} $$

$$(6 - 4 i)(6 + 4 i) = 52.\quad\halmos$$


10. Simplify $\dfrac{5 i}{6 - 4
   i}$ .

$$\dfrac{5 i}{6 - 4 i} = \dfrac{5 i}{6 - 4 i} \cdot \dfrac{6 + 4 i}{6 + 4 i} = \dfrac{5 i (6 + 4 i)}{(6 - 4 i)(6 + 4 i)} = \dfrac{30 i + 20 i^2}{6^2 - (4 i)^2} = \dfrac{30 i + 20 \cdot (-1)}{36 - 4^2 i^2} =$$

$$\dfrac{30 i - 20}{36 - 16 \cdot (-1)} = \dfrac{30 i - 20}{36 + 16} = \dfrac{30 i - 20}{52} = \dfrac{15 i - 10}{26}.\quad\halmos$$


11. Simplify $\dfrac{2 - 7 i}{4
   + 2 i}$ .

$$\dfrac{2 - 7 i}{4 + 2 i} = \dfrac{2 - 7 i}{4 + 2 i} \cdot \dfrac{4 - 2 i}{4 - 2 i} = \dfrac{(2 - 7 i)(4 - 2 i)}{(4 + 2 i)(4 - 2 i)} = \dfrac{-6 - 32 i}{20} = \dfrac{-3 - 16 i}{10}.$$

Here's the work for the multiplication on the top and bottom in the third step:

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & 2 & & $-7 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 4 & & 8 & & $-28 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $-2 i$ & & $-4 i$ & & $14 i^2 = -14$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} \hskip0.5in \vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & 4 & & $2 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 4 & & 16 & & $8 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $-2 i$ & & $-8 i$ & & $-4 i^2 = 4$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}\quad\halmos $$


12. Simplify the complex fraction $\dfrac{1 + 7 i}{3 - i}$ .

$$\dfrac{1 + 7 i}{3 - i} = \dfrac{1 + 7 i}{3 - i} \cdot \dfrac{3 + i}{3 + i} = \dfrac{3 + 21 i + i - 7}{9 + 1} =$$

$$\dfrac{-4 + 22 i}{10} = \dfrac{-2 + 11 i}{5}.$$

Here's the work for the multiplication on the top and the bottom:

$$\vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & 3 & & i & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 3 & & 9 & & $3 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & $-i$ & & $-3 i$ & & $-i^2 = 1$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }} \hskip0.5in \vbox{\offinterlineskip \halign{& \vrule # & \strut \hfil \quad # \quad \hfil \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & & & 1 & & $7 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & 3 & & 3 & & $21 i$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} height2pt & \omit & & \omit & & \omit & \cr & i & & i & & $7 i^2 = -7$ & \cr height2pt & \omit & & \omit & & \omit & \cr \noalign{\hrule} }}\quad\halmos $$


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