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Övningar 3.1

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Versionen från 16 juli 2007 kl. 08.07 (redigera)
KTH.SE:u1zpa8nw (Diskussion | bidrag)
(Ny sida: __NOTOC__ ==&Ouml;vning 3.1:1== <div class="ovning"> Skriv i potensform <table width="100%" cellspacing="10px"> <tr align="left"> <td class="ntext">a)</td> <td class="ntext" width="25%">$\s...)
← Gå till föregående ändring		 Nuvarande version (17 juli 2007 kl. 09.41) (redigera) (ogör)
KTH.SE:u1zpa8nw (Diskussion | bidrag)

 
(En mellanliggande version visas inte.)
Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==&Ouml;vning 3.1:1==+'''&Ouml;vning 3.1:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Skriv i potensform Skriv i potensform
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{2}</td> <td class="ntext" width="25%">\sqrt{2}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{7^5}</td> <td class="ntext" width="25%">\sqrt{7^5}</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4</td> <td class="ntext" width="25%">\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{\sqrt{3}}</td> <td class="ntext" width="25%">\sqrt{\sqrt{3}}</td>
</tr> </tr>
Rad 18: Rad 18:
</div> </div>
-==&Ouml;vning 3.1:2==+'''&Ouml;vning 3.1:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{3^2}</td> <td class="ntext" width="25%">\sqrt{3^2}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{\left(-3\right)^2}</td> <td class="ntext" width="25%">\sqrt{\left(-3\right)^2}</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{-3^2}</td> <td class="ntext" width="25%">\sqrt{-3^2}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5</td> <td class="ntext" width="25%">\sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt{18}\cdot\sqrt{8}</td> <td class="ntext" width="25%">\sqrt{18}\cdot\sqrt{8}</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt[\scriptstyle3]{8}</td> <td class="ntext" width="25%">\sqrt[\scriptstyle3]{8}</td>
-<td class="ntext">g)</td>+<td class="ntext">g)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\sqrt[\scriptstyle3]{-125}</td> <td class="ntext" width="25%">\sqrt[\scriptstyle3]{-125}</td>
<td></td> <td></td>
Rad 46: Rad 46:
</div> </div>
-==&Ouml;vning 3.1:3==+'''&Ouml;vning 3.1:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\bigl(\sqrt{5}-\sqrt{2}\,\bigr)\bigl(\sqrt{5}+\sqrt{2}\,\bigr)</td> <td class="ntext" width="50%">\bigl(\sqrt{5}-\sqrt{2}\,\bigr)\bigl(\sqrt{5}+\sqrt{2}\,\bigr)</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{\sqrt{96}}{\sqrt{18}}</td> <td class="ntext" width="50%">\displaystyle \frac{\sqrt{96}}{\sqrt{18}}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt{16+\sqrt{16}}</td> <td class="ntext" width="50%">\sqrt{16+\sqrt{16}}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)</td> <td class="ntext" width="50%">\sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)</td>
</tr> </tr>
Rad 66: Rad 66:
</div> </div>
-==&Ouml;vning 3.1:4==+'''&Ouml;vning 3.1:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt{0{,}16}</td> <td class="ntext" width="50%">\sqrt{0{,}16}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt[\scriptstyle3]{0{,}027}</td> <td class="ntext" width="50%">\sqrt[\scriptstyle3]{0{,}027}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80}</td> <td class="ntext" width="50%">\sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}</td> <td class="ntext" width="50%">\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}</td>
<td class="ntext"></td> <td class="ntext"></td>
Rad 87: Rad 87:
</div> </div>
-==&Ouml;vning 3.1:5==+'''&Ouml;vning 3.1:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Skriv som ett uttryck utan rottecken i n&auml;mnaren. Skriv som ett uttryck utan rottecken i n&auml;mnaren.
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\displaystyle \frac{2}{\sqrt{12}}</td> <td class="ntext" width="25%">\displaystyle \frac{2}{\sqrt{12}}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}</td> <td class="ntext" width="25%">\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\displaystyle \frac{2}{3+\sqrt{7}}</td> <td class="ntext" width="25%">\displaystyle \frac{2}{3+\sqrt{7}}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}</td> <td class="ntext" width="25%">\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}</td>
</tr> </tr>
Rad 105: Rad 105:
</div> </div>
-==&Ouml;vning 3.1:6==+'''&Ouml;vning 3.1:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Skriv som ett uttryck utan rottecken i n&auml;mnaren. Skriv som ett uttryck utan rottecken i n&auml;mnaren.
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}</td> <td class="ntext" width="50%">\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{1}{\left(\sqrt{3}-2\right)^2-2}</td> <td class="ntext" width="50%">\displaystyle \frac{1}{\left(\sqrt{3}-2\right)^2-2}</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{\displaystyle \frac{1}{\sqrt{3}}-\displaystyle \frac{1}{\sqrt{5}}}{\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{1}{2}}</td> <td class="ntext" width="50%">\displaystyle \frac{\displaystyle \frac{1}{\sqrt{3}}-\displaystyle \frac{1}{\sqrt{5}}}{\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{1}{2}}</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}</td> <td class="ntext" width="50%">\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}</td>
</tr> </tr>
Rad 125: Rad 125:
</div> </div>
-==&Ouml;vning 3.1:7==+'''&Ouml;vning 3.1:7'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt F&ouml;renkla s&aring; l&aring;ngt som m&ouml;jligt
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{1}{\sqrt{6}-\sqrt{5}} - \displaystyle \frac{1}{\sqrt{7}-\sqrt{6}}</td> <td class="ntext" width="33%">\displaystyle \frac{1}{\sqrt{6}-\sqrt{5}} - \displaystyle \frac{1}{\sqrt{7}-\sqrt{6}}</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}</td> <td class="ntext" width="33%">\displaystyle \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">\displaystyle \sqrt{153}-\sqrt{68}</td> <td class="ntext" width="33%">\displaystyle \sqrt{153}-\sqrt{68}</td>
</tr> </tr>
Rad 141: Rad 141:
</div> </div>
-==&Ouml;vning 3.1:8==+'''&Ouml;vning 3.1:8'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Avg&ouml;r vilket tal som &auml;r st&ouml;rst av Avg&ouml;r vilket tal som &auml;r st&ouml;rst av
-<table width="100%" cellspacing="10px">+<table width="100%">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt[\scriptstyle3]5\ och \ \sqrt[\scriptstyle3]6</td> <td class="ntext" width="50%">\sqrt[\scriptstyle3]5\ och \ \sqrt[\scriptstyle3]6</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt7\ och \ 7</td> <td class="ntext" width="50%">\sqrt7\ och \ 7</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt7\ och \ 2{,}5</td> <td class="ntext" width="50%">\sqrt7\ och \ 2{,}5</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">\sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\ och \ \sqrt[\scriptstyle3]2\cdot3</td> <td class="ntext" width="50%">\sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\ och \ \sqrt[\scriptstyle3]2\cdot3</td>
</tr> </tr>
Rad 160: Rad 160:
</table> </table>
</div> </div>
 +<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
 +<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>

Nuvarande version

Övning 3.1:1

Skriv i potensform

a)    \sqrt{2} b)    \sqrt{7^5} c)    \bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4 d)    \sqrt{\sqrt{3}}

Övning 3.1:2

Förenkla så långt som möjligt

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

Övning 3.1:3

Förenkla så långt som möjligt

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

Övning 3.1:4

Förenkla så långt som möjligt

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

Övning 3.1:5

Skriv som ett uttryck utan rottecken i nämnaren.

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

Övning 3.1:6

Skriv som ett uttryck utan rottecken i nämnaren.

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

Övning 3.1:7

Förenkla så långt som möjligt

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

Övning 3.1:8

Avgör vilket tal som är störst av

a)    \sqrt[\scriptstyle3]5\ och \ \sqrt[\scriptstyle3]6 b)    \sqrt7\ och \ 7
c)    \sqrt7\ och \ 2{,}5 d)    \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\ och \ \sqrt[\scriptstyle3]2\cdot3



















































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