3.1 Exercises

From Förberedande kurs i matematik 1

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{{Mall:Ej vald flik|[[3.1 Rötter|Teori]]}}
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{{Not selected tab|[[3.1 Roots|Theory]]}}
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{{Mall:Vald flik|[[3.1 Övningar|Övningar]]}}
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{{Selected tab|[[3.1 Exercises|Exercises]]}}
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===Övning 3.1:1===
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===Exercise 3.1:1===
<div class="ovning">
<div class="ovning">
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Skriv i potensform
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Write in power form
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="25%" | <math>\sqrt{\sqrt{3}}</math>
|width="25%" | <math>\sqrt{\sqrt{3}}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.1:1|Lösning a|Lösning 3.1:1a|Lösning b|Lösning 3.1:1b|Lösning c|Lösning 3.1:1c|Lösning d|Lösning 3.1:1d}}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:1|Solution a|Solution 3.1:1a|Solution b|Solution 3.1:1b|Solution c|Solution 3.1:1c|Solution d|Solution 3.1:1d}}
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===Övning 3.1:2===
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===Exercise 3.1:2===
<div class="ovning">
<div class="ovning">
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F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt
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Write in simplest possible form.
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="25%" | <math>\sqrt[\scriptstyle3]{-125}</math>
|width="25%" | <math>\sqrt[\scriptstyle3]{-125}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.1:2|Lösning a|Lösning 3.1:2a|Lösning b|Lösning 3.1:2b|Lösning c|Lösning 3.1:2c|Lösning d|Lösning 3.1:2d|Lösning 2|Lösning 3.1:2e|Lösning f|Lösning 3.1:2f|Lösning g|Lösning 3.1:2g}}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:2|Solution a|Solution 3.1:2a|Solution b|Solution 3.1:2b|Solution c|Solution 3.1:2c|Solution d|Solution 3.1:2d|Solution e|Solution 3.1:2e|Solution f|Solution 3.1:2f|Solution g|Solution 3.1:2g}}
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===Exercise 3.1:3===
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<div class="ovning">
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Write in simplest possible form.
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%" | <math>\bigl(\sqrt{5}-\sqrt{2}\,\bigr)\bigl(\sqrt{5}+\sqrt{2}\,\bigr)</math>
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|b)
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|width="50%" | <math>\displaystyle \frac{\sqrt{96}}{\sqrt{18}}</math>
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|-
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|c)
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|width="50%" | <math>\sqrt{16+\sqrt{16}}</math>
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|d)
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|width="50%" | <math>\sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:3|Solution a|Solution 3.1:3a|Solution b|Solution 3.1:3b|Solution c|Solution 3.1:3c|Solution d|Solution 3.1:3d}}
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===Exercise 3.1:4===
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<div class="ovning">
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Write in simplest possible form.
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%" | <math>\sqrt{0{,}16}</math>
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|b)
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|width="50%" | <math>\sqrt[\scriptstyle3]{0{,}027}</math>
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|-
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|c)
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|width="50%" | <math>\sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80}</math>
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|d)
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|width="50%" | <math>\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:4|Solution a|Solution 3.1:4a|Solution b|Solution 3.1:4b|Solution c|Solution 3.1:4c|Solution d|Solution 3.1:4d}}
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===Exercise 3.1:5===
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<div class="ovning">
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Write as an expression without a root sign in the denominator.
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{| width="100%" cellspacing="10px"
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|a)
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|width="25%" | <math>\displaystyle \frac{2}{\sqrt{12}}</math>
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|b)
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|width="25%" | <math>\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}</math>
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|c)
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|width="25%" | <math>\displaystyle \frac{2}{3+\sqrt{7}}</math>
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|d)
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|width="25%" | <math>\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:5|Solution a|Solution 3.1:5a|Solution b|Solution 3.1:5b|Solution c|Solution 3.1:5c|Solution d|Solution 3.1:5d}}
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===Exercise 3.1:6===
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<div class="ovning">
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Write as an expression without a root sign in the denominator.
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%" | <math>\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}</math>
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|b)
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|width="50%" | <math>\displaystyle \frac{1}{\left(\sqrt{3}-2\right)^2-2}</math>
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|-
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|c)
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|width="50%" | <math>\displaystyle \frac{\displaystyle \frac{1}{\sqrt{3}}-\displaystyle \frac{1}{\sqrt{5}}}{\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{1}{2}}</math>
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|d)
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|width="50%" | <math>\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:6|Solution a|Solution 3.1:6a|Solution b|Solution 3.1:6b|Solution c|Solution 3.1:6c|Solution d|Solution 3.1:6d}}
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===Exercise 3.1:7===
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<div class="ovning">
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Write in simplest possible form.
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{| width="100%" cellspacing="10px"
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|a)
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|width="33%" | <math>\displaystyle \frac{1}{\sqrt{6}-\sqrt{5}} - \displaystyle \frac{1}{\sqrt{7}-\sqrt{6}}</math>
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|b)
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|width="33%" | <math>\displaystyle \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}</math>
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|c)
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|width="33%" | <math>\displaystyle \sqrt{153}-\sqrt{68}</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:7|Solution a|Solution 3.1:7a|Solution b|Solution 3.1:7b|Solution c|Solution 3.1:7c}}
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===Exercise 3.1:8===
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<div class="ovning">
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Determine which number is the larger:
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%" | <math>\sqrt[\scriptstyle3]5\ </math> or <math>\ \sqrt[\scriptstyle3]6</math>
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|b)
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|width="50%" | <math>\sqrt7\ </math> or <math>\ 7</math>
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|-
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|c)
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|width="50%" | <math>\sqrt7\ </math> or <math>\ 2{.}5</math>
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|d)
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|width="50%" | <math>\sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\</math> or <math>\ \sqrt[\scriptstyle3]2\cdot3</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.1:8|Solution a|Solution 3.1:8a|Solution b|Solution 3.1:8b|Solution c|Solution 3.1:8c|Solution d|Solution 3.1:8d}}

Current revision

       Theory          Exercises      


Exercise 3.1:1

Write in power form

a) \displaystyle \sqrt{2} b) \displaystyle \sqrt{7^5} c) \displaystyle \bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4 d) \displaystyle \sqrt{\sqrt{3}}
Answer
Answer 3.1:1
Solution a
Solution 3.1:1a
Solution b
Solution 3.1:1b
Solution c
Solution 3.1:1c
Solution d
Solution 3.1:1d

Exercise 3.1:2

Write in simplest possible form.

a) \displaystyle \sqrt{3^2} b) \displaystyle \sqrt{\left(-3\right)^2} c) \displaystyle \sqrt{-3^2} d) \displaystyle \sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5
e) \displaystyle \sqrt{18}\cdot\sqrt{8} f) \displaystyle \sqrt[\scriptstyle3]{8} g) \displaystyle \sqrt[\scriptstyle3]{-125}
Answer
Answer 3.1:2
Solution a
Solution 3.1:2a
Solution b
Solution 3.1:2b
Solution c
Solution 3.1:2c
Solution d
Solution 3.1:2d
Solution e
Solution 3.1:2e
Solution f
Solution 3.1:2f
Solution g
Solution 3.1:2g

Exercise 3.1:3

Write in simplest possible form.

a) \displaystyle \bigl(\sqrt{5}-\sqrt{2}\,\bigr)\bigl(\sqrt{5}+\sqrt{2}\,\bigr) b) \displaystyle \displaystyle \frac{\sqrt{96}}{\sqrt{18}}
c) \displaystyle \sqrt{16+\sqrt{16}} d) \displaystyle \sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)
Answer
Answer 3.1:3
Solution a
Solution 3.1:3a
Solution b
Solution 3.1:3b
Solution c
Solution 3.1:3c
Solution d
Solution 3.1:3d

Exercise 3.1:4

Write in simplest possible form.

a) \displaystyle \sqrt{0{,}16} b) \displaystyle \sqrt[\scriptstyle3]{0{,}027}
c) \displaystyle \sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80} d) \displaystyle \sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}
Answer
Answer 3.1:4
Solution a
Solution 3.1:4a
Solution b
Solution 3.1:4b
Solution c
Solution 3.1:4c
Solution d
Solution 3.1:4d

Exercise 3.1:5

Write as an expression without a root sign in the denominator.

a) \displaystyle \displaystyle \frac{2}{\sqrt{12}} b) \displaystyle \displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}} c) \displaystyle \displaystyle \frac{2}{3+\sqrt{7}} d) \displaystyle \displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}
Answer
Answer 3.1:5
Solution a
Solution 3.1:5a
Solution b
Solution 3.1:5b
Solution c
Solution 3.1:5c
Solution d
Solution 3.1:5d

Exercise 3.1:6

Write as an expression without a root sign in the denominator.

a) \displaystyle \displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2} b) \displaystyle \displaystyle \frac{1}{\left(\sqrt{3}-2\right)^2-2}
c) \displaystyle \displaystyle \frac{\displaystyle \frac{1}{\sqrt{3}}-\displaystyle \frac{1}{\sqrt{5}}}{\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{1}{2}} d) \displaystyle \displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}
Answer
Answer 3.1:6
Solution a
Solution 3.1:6a
Solution b
Solution 3.1:6b
Solution c
Solution 3.1:6c
Solution d
Solution 3.1:6d

Exercise 3.1:7

Write in simplest possible form.

a) \displaystyle \displaystyle \frac{1}{\sqrt{6}-\sqrt{5}} - \displaystyle \frac{1}{\sqrt{7}-\sqrt{6}} b) \displaystyle \displaystyle \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}} c) \displaystyle \displaystyle \sqrt{153}-\sqrt{68}
Answer
Answer 3.1:7
Solution a
Solution 3.1:7a
Solution b
Solution 3.1:7b
Solution c
Solution 3.1:7c

Exercise 3.1:8

Determine which number is the larger:

a) \displaystyle \sqrt[\scriptstyle3]5\ or \displaystyle \ \sqrt[\scriptstyle3]6 b) \displaystyle \sqrt7\ or \displaystyle \ 7
c) \displaystyle \sqrt7\ or \displaystyle \ 2{.}5 d) \displaystyle \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\ or \displaystyle \ \sqrt[\scriptstyle3]2\cdot3
Answer
Answer 3.1:8
Solution a
Solution 3.1:8a
Solution b
Solution 3.1:8b
Solution c
Solution 3.1:8c
Solution d
Solution 3.1:8d
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