3.3 Exercises

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{{Mall:Ej vald flik|[[3.3 Potenser och rötter|Teori]]}}
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{{Not selected tab|[[3.3 Powers and roots|Theory]]}}
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{{Mall:Vald flik|[[3.3 Övningar|Övningar]]}}
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{{Selected tab|[[3.3 Exercises|Exercises]]}}
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===Övning 3.3:1===
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===Exercise 3.3:1===
<div class="ovning">
<div class="ovning">
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Skriv följande tal i formen <math>\,a+ib\,</math>, där <math>\,a\,</math> och <math>\,b\,</math> är reella tal.
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Write the following number in the form <math>\,a+ib\,</math>, where <math>\,a\,</math> and <math>\,b\,</math> are real numbers:
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%"| <math>\displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}</math>
|width="50%"| <math>\displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:1|Lösning a|Lösning 3.3:1a|Lösning b|Lösning 3.3:1b|Lösning c|Lösning 3.3:1c|Lösning d|Lösning 3.3:1d|Lösning e|Lösning 3.3:1e}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:1|Solution a|Solution 3.3:1a|Solution b|Solution 3.3:1b|Solution c|Solution 3.3:1c|Solution d|Solution 3.3:1d|Solution e|Solution 3.3:1e}}
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===Övning 3.3:2===
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===Exercise 3.3:2===
<div class="ovning">
<div class="ovning">
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Lös ekvationerna
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Solve the equations
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
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|a)
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|width="33%"| <math>\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1</math>
|width="33%"| <math>\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:2|Lösning a|Lösning 3.3:2a|Lösning b|Lösning 3.3:2b|Lösning c|Lösning 3.3:2c|Lösning d|Lösning 3.3:2d|Lösning e|Lösning 3.3:2e}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:2|Solution a|Solution 3.3:2a|Solution b|Solution 3.3:2b|Solution c|Solution 3.3:2c|Solution d|Solution 3.3:2d|Solution e|Solution 3.3:2e}}
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===Övning 3.3:3===
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===Exercise 3.3:3===
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<div class="ovning">
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Kvadratkomplettera följande uttryck
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Complete the square of the following expressions
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%"| <math>iz^2+(2+3i)z-1</math>
|width="50%"| <math>iz^2+(2+3i)z-1</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:3|Lösning a|Lösning 3.3:3a|Lösning b|Lösning 3.3:3b|Lösning c|Lösning 3.3:3c|Lösning d|Lösning 3.3:3d}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:3|Solution a|Solution 3.3:3a|Solution b|Solution 3.3:3b|Solution c|Solution 3.3:3c|Solution d|Solution 3.3:3d}}
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===Övning 3.3:4===
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===Exercise 3.3:4===
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<div class="ovning">
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Lös ekvationerna
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Solve the equations
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{| width="100%" cellspacing="10px"
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|a)
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|c)
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|width="50%"| <math>-z^2+2z+3=0</math>
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|width="50%"| <math>z^2+2z+3=0</math>
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|d)
|width="50%"| <math>\displaystyle\frac{1}{z} + z = \frac{1}{2}</math>
|width="50%"| <math>\displaystyle\frac{1}{z} + z = \frac{1}{2}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:4|Lösning a|Lösning 3.3:4a|Lösning b|Lösning 3.3:4b|Lösning c|Lösning 3.3:4c|Lösning d|Lösning 3.3:4d}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:4|Solution a|Solution 3.3:4a|Solution b|Solution 3.3:4b|Solution c|Solution 3.3:4c|Solution d|Solution 3.3:4d}}
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===Övning 3.3:5===
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===Exercise 3.3:5===
<div class="ovning">
<div class="ovning">
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Lös ekvationerna
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Solve the equations
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%"| <math>(4+i)z^2+(1-21i)z=17</math>
|width="50%"| <math>(4+i)z^2+(1-21i)z=17</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:5|Lösning a|Lösning 3.3:5a|Lösning b|Lösning 3.3:5b|Lösning c|Lösning 3.3:5c|Lösning d|Lösning 3.3:5d}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:5|Solution a|Solution 3.3:5a|Solution b|Solution 3.3:5b|Solution c|Solution 3.3:5c|Solution d|Solution 3.3:5d}}
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===Exercise 3.3:6===
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<div class="ovning">
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Determine the solution to <math>\,z^2=1+i\,</math> both in polar form and in the form <math>\,a+ib\,</math>, where <math>\,a\,</math> and <math>\,b\,</math> are real numbers. Use the result to calculate <math>\; \tan \frac{\pi}{8}\,</math>.
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</div>{{#NAVCONTENT:Answer|Answer 3.3:6|Solution|Solution 3.3:6}}

Current revision

       Theory          Exercises      

Exercise 3.3:1

Write the following number in the form \displaystyle \,a+ib\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers:

a) \displaystyle (i+1)^{12} b) \displaystyle \displaystyle\Bigl(\frac{1+i\sqrt{3}}{2}\,\Bigr)^{12}
c) \displaystyle (4\sqrt{3} -4i)^{22} d) \displaystyle \Bigl(\displaystyle\frac{1+i\sqrt{3}}{1+i}\,\Bigr)^{12}
e) \displaystyle \displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9}
Answer
Answer 3.3:1
Solution a
Solution 3.3:1a
Solution b
Solution 3.3:1b
Solution c
Solution 3.3:1c
Solution d
Solution 3.3:1d
Solution e
Solution 3.3:1e

Exercise 3.3:2

Solve the equations

a) \displaystyle z^4=1 b) \displaystyle z^3=-1 c) \displaystyle z^5=-1-i
d) \displaystyle (z-1)^4+4=0 e) \displaystyle \displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1
Answer
Answer 3.3:2
Solution a
Solution 3.3:2a
Solution b
Solution 3.3:2b
Solution c
Solution 3.3:2c
Solution d
Solution 3.3:2d
Solution e
Solution 3.3:2e

Exercise 3.3:3

Complete the square of the following expressions

a) \displaystyle z^2 +2z+3 b) \displaystyle z^2 +3iz-\frac{1}{4}
c) \displaystyle -z^2-2iz +4z+1 d) \displaystyle iz^2+(2+3i)z-1
Answer
Answer 3.3:3
Solution a
Solution 3.3:3a
Solution b
Solution 3.3:3b
Solution c
Solution 3.3:3c
Solution d
Solution 3.3:3d

Exercise 3.3:4

Solve the equations

a) \displaystyle z^2=i b) \displaystyle z^2-4z+5=0
c) \displaystyle z^2+2z+3=0 d) \displaystyle \displaystyle\frac{1}{z} + z = \frac{1}{2}
Answer
Answer 3.3:4
Solution a
Solution 3.3:4a
Solution b
Solution 3.3:4b
Solution c
Solution 3.3:4c
Solution d
Solution 3.3:4d

Exercise 3.3:5

Solve the equations

a) \displaystyle z^2-2(1+i)z+2i-1=0 b) \displaystyle z^2-(2-i)z+(3-i)=0
c) \displaystyle z^2-(1+3i)z-4+3i=0 d) \displaystyle (4+i)z^2+(1-21i)z=17
Answer
Answer 3.3:5
Solution a
Solution 3.3:5a
Solution b
Solution 3.3:5b
Solution c
Solution 3.3:5c
Solution d
Solution 3.3:5d

Exercise 3.3:6

Determine the solution to \displaystyle \,z^2=1+i\, both in polar form and in the form \displaystyle \,a+ib\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers. Use the result to calculate \displaystyle \; \tan \frac{\pi}{8}\,.

Answer
Answer 3.3:6
Solution
Solution 3.3:6
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