# 21.3: Exercises

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## Exercise $$\PageIndex{1}$$

Plot the complex numbers in the complex plane.

1. $$4+2i$$
2. $$-3-5i$$
3. $$6-2i$$
4. $$-5+i$$
5. $$-2i$$
6. $$\sqrt{2}-\sqrt{2}i$$
7. $$7$$
8. $$i$$
9. $$0$$
10. $$2i-\sqrt{3}$$

## Exercise $$\PageIndex{2}$$

Add, subtract, multiply, and divide, as indicated.

1. $$(5-2i)+(-2+6i)$$
2. $$(-9-i)-(5-3i)$$
3. $$(3+2i)\cdot (4+3i)$$
4. $$(-2-i)\cdot (-1+4i)$$
5. $$\dfrac{2+3i}{2+i}$$
6. $$(5+5i)\div (2-4i)$$
1. $$3+4 i$$
2. $$-14+2 i$$
3. $$6+17 i$$
4. $$6-7 i$$
5. $$\dfrac{7}{5}+\dfrac{4}{5} i$$
6. $$-\dfrac{1}{2}+\dfrac{3}{2} i$$

## Exercise $$\PageIndex{3}$$

Find the absolute value $$|a+bi|$$ of the given complex number, and simplify your answer as much as possible.

1. $$|4+3i|$$
2. $$|1-2i|$$
3. $$|-3i|$$
4. $$|-2-6i|$$
5. $$|\sqrt{8}-i|$$
6. $$|-2\sqrt{3}-2i|$$
7. $$|-5|$$
8. $$|-\sqrt{17}+4\sqrt{2}i|$$
1. $$5$$
2. $$\sqrt{5}$$
3. $$3$$
4. $$2 \sqrt{10}$$
5. $$3$$
6. $$4$$
7. $$5$$
8. $$7$$

## Exercise $$\PageIndex{4}$$

Convert the complex number into polar form $$r(\cos(\theta)+i\sin(\theta))$$.

1. $$2+2i$$
2. $$4\sqrt{3}+4i$$
3. $$3-2i$$
4. $$-5+5i$$
5. $$4-3i$$
6. $$-4+3i$$
7. $$-\sqrt{5}-\sqrt{15}i$$
8. $$\sqrt{7}-\sqrt{21}i$$
9. $$-5-12i$$
10. $$6i$$
11. $$-10$$
12. $$-\sqrt{3}+3i$$
1. $$2 \sqrt{2}\left(\cos \left(\dfrac{\pi}{4}\right)+i \sin \left(\dfrac{\pi}{4}\right)\right)$$
2. $$8\left(\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)\right)$$
3. approximately $$\sqrt{13}(\cos (-.588)+i \sin (-.588))$$ or $$\sqrt{13}(\cos (-.187 \pi)+i \sin (-.187 \pi))$$
4. $$5 \sqrt{2}\left(\cos \left(\dfrac{3 \pi}{4}\right)+i \sin \left(\dfrac{3 \pi}{4}\right)\right)$$
5. approximately $$5(\cos (-.644)+i \sin (-.644))$$ or $$5(\cos (-.205 \pi)+i \sin (-.205 \pi))$$
6. approximately $$5(\cos (2.498)+i \sin (2.498))$$ or $$5(\cos (.795 \pi)+i \sin (.795 \pi))$$
7. $$2 \sqrt{5}\left(\cos \left(\dfrac{4 \pi}{3}\right)+i \sin \left(\dfrac{4 \pi}{3}\right)\right)$$
8. $$2 \sqrt{7}\left(\cos \left(-\dfrac{\pi}{3}\right)+i \sin \left(-\dfrac{\pi}{3}\right)\right)$$
9. approximately $$13(\cos (4.318)+i \sin (4.318))$$ or $$13(\cos (1.374 \pi)+i \sin (1.374 \pi))$$
10. $$6\left(\cos \left(\dfrac{\pi}{2}\right)+i \sin \left(\dfrac{\pi}{2}\right)\right)$$
11. $$10(\cos (\pi)+i \sin (\pi))$$
12. $$2 \sqrt{3}\left(\cos \left(\dfrac{2 \pi}{3}\right)+ i \sin \left(\dfrac{2 \pi}{3}\right)\right)$$

## Exercise $$\PageIndex{5}$$

Convert the complex number into the standard form $$a+bi$$.

1. $$6(\cos(134^\circ)+i\sin(134^\circ))$$
2. $$\dfrac 1 2 \left(\cos\left(\dfrac \pi {17}\right)+i\sin\left(\dfrac \pi {17}\right)\right)$$
3. $$2(\cos(270^\circ)+i\sin(270^\circ))$$
4. $$\cos\left(\dfrac{\pi} 6\right)+i\sin\left(\dfrac{\pi}6\right)$$
5. $$10\left(\cos\left(\dfrac{7\pi}{6}\right)+i\sin\left(\dfrac{7\pi}{6}\right)\right)$$
6. $$6 \left(\cos\left(-\dfrac{5\pi}{12}\right)+i\sin\left(-\dfrac{5\pi}{12}\right)\right)$$
1. approximately $$-4.168+4.316 i$$
2. approximately $$.491+0.0919 i$$
3. $$-2 i$$
4. $$\dfrac{\sqrt{3}}{2}+\dfrac{1}{2} i$$
5. $$-5 \sqrt{3}-5 i$$
6. approximately $$1.553-5 / 796 i$$

## Exercise $$\PageIndex{6}$$

Multiply the complex numbers and write the answer in standard form $$a+bi$$.

1. $$4(\cos(27^\circ)+i\sin(27^\circ)) \cdot 10(\cos(33^\circ)+i\sin(33^\circ))$$
2. $$7\left(\cos\left(\dfrac{2\pi}{9}\right)+i\sin\left(\dfrac{2\pi}{9}\right)\right) \cdot 6\left(\cos\left(\dfrac{\pi}{9}\right)+i\sin\left(\dfrac{\pi}{9}\right)\right)$$
3. $$\left(\cos\left(\dfrac{13\pi}{12}\right)+i\sin\left(\dfrac{13\pi}{12}\right)\right) \cdot \left(\cos\left(\dfrac{-11\pi}{12}\right)+i\sin\left(\dfrac{-11\pi}{12}\right)\right)$$
4. $$8\left(\cos\left(\dfrac{3\pi}{7}\right)+i\sin\left(\dfrac{3\pi}{7}\right)\right) \cdot 1.5\left(\cos\left(\dfrac{4\pi}{7}\right)+i\sin\left(\dfrac{4\pi}{7}\right)\right)$$
5. $$0.2(\cos(196^\circ)+i\sin(196^\circ)) \cdot 0.5(\cos(88^\circ)+i\sin(88^\circ))$$
6. $$4\left(\cos\left(\dfrac{7\pi}{8}\right)+i\sin\left(\dfrac{7\pi}{8}\right)\right) \cdot 0.25\left(\cos\left(\dfrac{-5\pi}{24}\right)+i\sin\left(\dfrac{-5\pi}{24}\right)\right)$$
1. $$40\left(\cos \left(60^{\circ}\right)+i \sin \left(60^{\circ}\right)\right)=20+20 \sqrt{3} i$$
2. $$42\left(\cos \left(\dfrac{\pi}{3}\right)+i \sin \left(\dfrac{\pi}{3}\right) \right)=21+21 \sqrt{3} i$$
3. $$\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2} i$$
4. $$12(\cos (\pi)+i \sin (\pi))=-12$$
5. $$.1\left(\cos \left(284^{\circ}\right)+i \sin \left(284^{\circ}\right)\right) \approx .0242-.0970 i$$
6. $$\cos \left(\dfrac{2 \pi}{3}\right)+i \sin \left(\dfrac{2 \pi}{3}\right)=-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2} i$$

## Exercise $$\PageIndex{7}$$

Divide the complex numbers and write the answer in standard form $$a+bi$$.

1. $$\displaystyle\frac{18(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))}{3(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))}$$
2. $$\displaystyle\frac{10(\cos(254^\circ)+i\sin(254^\circ))}{15(\cos(164^\circ)+i\sin(164^\circ))}$$
3. $$\displaystyle\frac{\sqrt{24}(\cos(\frac{11\pi}{14})+i\sin(\frac{11\pi}{14}))}{\sqrt{6}(\cos(\frac{2\pi}{7})+i\sin(\frac{2\pi}{7}))}$$
4. $$\displaystyle\frac{\cos(\frac{8\pi}{5})+i\sin(\frac{8\pi}{5})}{2(\cos(\frac{\pi}{10})+i\sin(\frac{\pi}{10}))}$$
5. $$\displaystyle\frac{42(\cos(\frac{7\pi}{4})+i\sin(\frac{7\pi}{4}))}{7(\cos(\frac{5\pi}{12})+i\sin(\frac{5\pi}{12}))}$$
6. $$\displaystyle\frac{30(\cos(-175^\circ)+i\sin(-175^\circ))}{18(\cos(144^\circ)+i\sin(144^\circ))}$$
1. $$6(\cos (\pi / 3)+i \sin (\pi / 3))=3+3 \sqrt{3} i$$
2. $$\dfrac{2}{3}\left(\cos \left(90^{\circ}\right)+i \sin \left(90^{\circ}\right)\right)=\dfrac{2}{3} i$$
3. $$2(\cos (\pi / 2)+i \sin (\pi / 2))=2 i$$
4. $$\dfrac{1}{2}\left(\cos \left(\dfrac{3 \pi}{2}\right)+i \sin \left(\dfrac{3 \pi}{2}\right)\right)=-\dfrac{1}{2} i$$
5. $$6\left(\cos \left(\dfrac{4 \pi}{3}\right)+i \sin \left(\dfrac{4 \pi}{3}\right)\right)=-3-3 \sqrt{3} i$$
6. $$\dfrac{5}{3}\left(\cos \left(-319^{\circ}\right)+i \sin \left(-319^{\circ}\right)\right) \approx 1.258+1.093 i$$