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Sammanställning övningar

Sommarmatte 1

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(→Övning 2.1:5)
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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==Övning 1.1:1==+[[övn 1|1 Numerisk räkning]]
-<div class="ovning">+
-Beräkna (utan hjälp av räknedosa)+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">3-7-4+6-5</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">3-(7-4)+(6-5)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">3-(7-(4+6)-5)</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">3-(7-(4+6))-5</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
-<div class="svar">+[[övn 2|2 Algebra]]
-<table width="100%" cellspacing="10px">+[[övn 3|3 Rötter och algoritmer]]
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">-7</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">1</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">11</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">1</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+[[övn 4|4 Trigonometri]]
- +
-==Övning 1.1:2==+
-<div class="ovning">+
-Beräkna (utan hjälp av räknedosa)+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">(3-(7-4))(6-5)</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">3-(((7-4)+6)-5)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">3\cdot(-7)-4\cdot(6-5)</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">3\cdot(-7)-(4+6)/(-5)</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">0</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">-1</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">-25</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">-19</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.1:3==+
-<div class="ovning">+
-Vilka av följande tal tillhör de naturliga talen? heltalen?+
-rationella talen? irrationella talen? Förenkla först!+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">8</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">-4</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">8-4</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">4-8</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">8(-4)</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">(-8)(-4)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">g)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{4}{-8}</td>+
-<td class="ntext">h)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{-8}{-4}</td>+
-<td class="ntext">i)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{\sqrt{2}}{3}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">j)</td>+
-<td class="ntext" width="33%">\displaystyle \Bigl(\frac{4}{\sqrt{2}}\Bigr)^2</td>+
-<td class="ntext">k)</td>+
-<td class="ntext" width="33%">-\pi</td>+
-<td class="ntext">l)</td>+
-<td class="ntext" width="33%">\pi+1</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr><td height="5px"/></tr>+
-<tr align="left" valign="top">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">naturliga talen, heltalen, rationella talen</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">heltalen, rationella talen</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">naturliga talen, heltalen, rationella talen</td>+
-</tr>+
-<tr align="left" valign="top">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">heltalen, rationella talen</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">heltalen, rationella talen</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">naturliga talen, heltalen, rationella talen</td>+
-</tr>+
-<tr align="left" valign="top">+
-<td class="ntext">g)</td>+
-<td class="ntext" width="33%">rationella talen</td>+
-<td class="ntext">h)</td>+
-<td class="ntext" width="33%">naturliga talen, heltalen, rationella talen</td>+
-<td class="ntext">i)</td>+
-<td class="ntext" width="33%">irrationella talen</td>+
-</tr>+
-<tr align="left" valign="top">+
-<td class="ntext">j)</td>+
-<td class="ntext" width="33%">naturliga talen, heltalen, rationella talen</td>+
-<td class="ntext">k)</td>+
-<td class="ntext" width="33%">irrationella talen</td>+
-<td class="ntext">l)</td>+
-<td class="ntext" width="33%">irrationella talen</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.1:4==+
-<div class="ovning">+
-Ordna följande tal i storleksordning+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="100%">\displaystyle 2,\ \frac{3}{5},\ \frac{5}{3}\ och \ \displaystyle \frac{7}{3}</td>+
-</tr>+
-<tr>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="100%">\displaystyle -\frac{1}{2},\ -\frac{1}{5},\ -\frac{3}{10}\ och \ \displaystyle -\frac{1}{3}</td>+
-</tr>+
-<tr>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="100%">\displaystyle \frac{1}{2},\ \frac{2}{3},\ \frac{3}{5},\ \frac{5}{8}\ och \ \displaystyle \frac{21}{34}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="100%">\displaystyle \frac{3}{5}<\frac{5}{3}<2<\frac{7}{3}</td>+
-</tr>+
-<tr>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="100%">\displaystyle -\frac{1}{2}<-\frac{1}{3}<-\frac{3}{10}<-\frac{1}{5}</td>+
-</tr>+
-<tr>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="100%">\displaystyle \frac{1}{2}<\frac{3}{5}<\frac{21}{34}<\frac{5}{8}<\frac{2}{3}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.1:5==+
-<div class="ovning">+
-Ange decimalutvecklingen med tre korrekta decimaler till+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="25%">\displaystyle \frac{7}{6}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="25%">\displaystyle \frac{9}{4}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="25%">\displaystyle \frac{2}{7}</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="25%">\sqrt{2}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="25%">1{,}167</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="25%">2{,}250</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="25%">0{,}286</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="25%">1{,}414</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.1:6==+
-<div class="ovning">+
-Vilka av följande tal är rationella? Ange dem som en kvot mellan heltal.+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="100%">3,14</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">b)</td>+
-<td class="ntext" width="100%">3{,}1416\,1416\,1416\,\dots</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="100%">0{,}2\,001\,001\,001\,\dots\, (därefter är var tredje decimal en 1:a och övriga 0)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="100%">0{,}10\,100\,1000\,10000\,1\dots\, (en 1:a, en 0:a, en 1:a, två 0:or, en 1:a, tre 0:or osv.)</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="100%">Talet är rationellt och lika med \,314/100 = 157/50\,.</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">b)</td>+
-<td class="ntext" width="100%">Talet är rationellt och är lika med \,31413/9999 = 10471/3333\,.</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="100%">Talet är rationellt och lika med \,1999/9990\,.</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="100%">Talet är irrationellt.</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
- +
-==Övning 1.2:1==+
-<div class="ovning">+
-Skriv p&aring; gemensamt br&aring;kstreck +
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{7}{4}+\frac{11}{7}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{2}{7}-\frac{1}{5}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{1}{6}-\frac{2}{5}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{1}{3}+\frac{1}{4}+\frac{1}{5}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{8}{7}+\frac{3}{4}-\frac{4}{3}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{93}{28}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{3}{35}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle -\frac{7}{30}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{47}{60}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{47}{84}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.2:2==+
-<div class="ovning">+
-Bestäm minsta gemensamma nämnare+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{6}+\frac{1}{10}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{4}-\frac{1}{8}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{12}-\frac{1}{14}</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{2}{45}+\frac{1}{75}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\displaystyle {30}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle {8}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">\displaystyle {84}</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">\displaystyle {225}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.2:3==+
-<div class="ovning">+
-Beräkna följande uttryck genom att använda minsta gemensamma nämnare:+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{3}{20}+\frac{7}{50}-\frac{1}{10}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{24}+\frac{1}{40}-\frac{1}{16}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr><td height="5px"/></tr>+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{19}{100}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{240}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.2:4==+
-<div class="ovning">+
-F&ouml;renkla f&ouml;ljande uttryck genom att skriva p&aring; gemensamt br&aring;kstreck. Br&aring;ket ska vara f&auml;rdigf&ouml;rkortat.+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle\frac{\displaystyle\frac{3}{5}}{\displaystyle\frac{7}{10}}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle\frac{\displaystyle\frac{2}{7}}{\displaystyle\frac{3}{8}}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle\frac{\displaystyle\frac{1}{4}-\frac{1}{5}}{\displaystyle\frac{3}{10}}</td>+
-</tr><tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{6}{7}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{16}{21}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{1}{6}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==&Ouml;vning 1.2:5==+
-<div class="ovning">+
-F&ouml;renkla f&ouml;ljande uttryck genom att skriva p&aring; gemensamt br&aring;kstreck. Br&aring;ket ska vara f&auml;rdigf&ouml;rkortat.+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{2}{\displaystyle \frac{1}{7}\displaystyle -\frac{1}{15}}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle\frac{\displaystyle\frac{1}{2}\displaystyle+\frac{1}{3}}{\displaystyle\frac{1}{3}\displaystyle-\frac{1}{2}}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle\frac{\displaystyle\frac{3}{10}\displaystyle-\frac{1}{5}}{\displaystyle\frac{7}{8}\displaystyle-\frac{3}{16}}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr><td height="5px"/></tr>+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{105}{4}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">-5</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{8}{55}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 1.2:6==+
- +
-<div class="ovning">+
-F&ouml;renkla \ \,\displaystyle \frac{\displaystyle \frac{2}{\displaystyle 3+\frac{1}{2}}\displaystyle + \frac{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{4}\displaystyle -\frac{1}{3}}}{\displaystyle \frac{1}{2}\displaystyle - \frac{3}{\displaystyle 2-\frac{2}{7}}}+
-</div>+
- +
-<div class="svar">+
-<table align="center" width="100%"><tr><td class="ntext" align="center">\displaystyle \frac{152}{35}</td></tr></table> +
-</div>+
- +
- +
-==&Ouml;vning 1.3:1==+
- +
-<div class="ovning">+
-Ber&auml;kna +
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="25%">2^3\cdot3^2</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="25%">3^5\cdot9^{-2}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="25%">(-5)^3</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="25%">\Bigl(\displaystyle \frac{2}{3}\Bigr)^{-3}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="25%">72</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="25%">3</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="25%">-125</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="25%">\displaystyle \frac{27}{8}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==&Ouml;vning 1.3:2==+
- +
-<div class="ovning">+
-Skriv som en potens av 2+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">2\cdot4\cdot8</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">0{,}25</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">1</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">2^6</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">2^{-2}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">2^0</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==&Ouml;vning 1.3:3==+
- +
-<div class="ovning">+
-Skriv som en potens av 3+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="20%">\displaystyle \frac{1}{3}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="20%">243</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="20%">9^2</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="20%">\displaystyle \frac{1}{27}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="20%">\displaystyle \frac{3}{9^2}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="20%">3^{-1}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="20%">3^5</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="20%">3^4</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="20%">3^{-3}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="20%">3^{-3}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==&Ouml;vning 1.3:4==+
- +
-<div class="ovning">+
-Ber&auml;kna+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">2^{9}\cdot2^{-7}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">3^{13}\cdot9^{-3}\cdot27^{\,-2}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{5^{12}}{5^{-4}}\cdot(5^{2})^{-6}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">2^{2^{\scriptstyle3}}\cdot(-2)^{\scriptstyle-4}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">625\cdot(5^{8}+5^{9})^{-1}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">4</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">3</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">625</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">16</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{1}{3750}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==&Ouml;vning 1.3:5==+
- +
-<div class="ovning">+
-Ber&auml;kna+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">4^{1/2}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">4^{-1/2}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">9^{3/2}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">\left(47^{2/3} \right) ^{3}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">3^{1{,}4}\cdot3^{0{,}6}</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">\bigl( 125 ^{1/3} \bigr)^2\cdot \bigl( 27^{1/3} \bigr)^{-2}\cdot9^{1/2}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">2</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{1}{2}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">27</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">2209</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">9</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{25}{3}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
- +
-==&Ouml;vning 1.3:6==+
-<div class="ovning">+
-Avg&ouml;r vilket tal som &auml;r st&ouml;rst av +
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">256^{1/3}\ och \ 200^{1/3}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">0{,}5^{-3}\ och \ 0{,}4^{-3}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">0{,}2^5\ och \ 0{,}2^{7}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">400^{1/3}\ och \ \bigl(5^{1/3}\bigr)^{4}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">125^{1/2}\ och \ 625^{1/3}</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">2^{56}\ och \ 3^{40}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">256^{1/3}>200^{1/3}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">0{,}4^{-3}>0{,}5^{-3}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">0{,}2^{5}>0{,}2^{7}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">\bigl(5^{1/3}\bigr)^{4}>400^{1/3}</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">125^{1/2}>625^{1/3}</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">3^{40}>2^{56}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 2.1:1==+
-<div class="ovning">+
-Utveckla+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">3x(x-1)</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">(1+x-x^2)xy</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">-x^2(4-y^2)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">x^3y^2\left(\displaystyle \frac{1}{y} - \frac{1}{xy}+1\right)</td>+
-<td class="ntext">e) </td>+
-<td class="ntext" width="33%">(x-7)^2</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">(5+4y)^2</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">g)</td>+
-<td class="ntext" width="33%">(y^2-3x^3)^2</td>+
-<td class="ntext">h)</td>+
-<td class="ntext" width="33%">(5x^3+3x^5)^2</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">3x^2-3x</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">xy+x^2y-x^3y</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">-4x^2+x^2y^2</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">x^3y-x^2y+x^3y^2</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">x^2-14x+49</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">16y^2+40y+25</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">g)</td>+
-<td class="ntext" width="33%">9x^6-6x^3y^2+y^4</td>+
-<td class="ntext">h)</td>+
-<td class="ntext" width="33%">9x^{10}+30x^8+25x^6</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 2.1:2==+
-<div class="ovning">+
-Utveckla och förenkla så långt som möjligt+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">(x-4)(x-5)-3x(2x-3)</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">(1-5x)(1+15x)-3(2-5x)(2+5x)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">(3x+4)^2-(3x-2)(3x-8)</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">(3x^2+2)(3x^2-2)(9x^4+4)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">e)</td>+
-<td class="ntext" width="50%">(a+b)^2+(a-b)^2</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">-5x^2+20</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">10x-11</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">54x</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">81x^8-16</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">e)</td>+
-<td class="ntext" width="50%">2a^2+2b^2</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 2.1:3==+
- +
-<div class="ovning">+
-Faktorisera s&aring; l&aring;ngt som m&ouml;jligt+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">x^2-36</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">5x^2-20</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">x^2+6x+9</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">x^2-10x+25</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">18x-2x^3</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">16x^2+8x+1</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">(x+6)(x-6)</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">5(x+2)(x-2)</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">(x+3)^2</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">d)</td>+
-<td class="ntext" width="33%">(x-5)^2</td>+
-<td class="ntext">e)</td>+
-<td class="ntext" width="33%">-2x(x+3)(x-3)</td>+
-<td class="ntext">f)</td>+
-<td class="ntext" width="33%">(4x+1)^2</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 2.1:4==+
-<div class="ovning">+
-Bestäm koefficienterna framför \,x\, och \,x^2\, när följande uttryck utvecklas+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="100%">(x+2)(3x^2-x+5)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">b)</td>+
-<td class="ntext" width="100%">(1+x+x^2+x^3)(2-x+x^2+x^4)</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="100%">(x-x^3+x^5)(1+3x+5x^2)(2-7x^2-x^4)</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="100%">5\, framf&ouml;r \,x^2\,, \,3\, framf&ouml;r \,x</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">b)</td>+
-<td class="ntext" width="100%">2\, framf&ouml;r \,x^2\,, \,1\, framf&ouml;r \,x</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">\textrm{c) }</td>+
-<td class="ntext" width="100%">6\, framf&ouml;r \,x^2\,, \,2\, framf&ouml;r \,x</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-==Övning 2.1:5==+
-<div class="ovning">+
-Förenkla så långt som möjligt+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{x-x^2}-\displaystyle \frac{1}{x}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{y^2-2y}-\displaystyle \frac{2}{y^2-4}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{(3x^2-12)(x^2-1)}{(x+1)(x+2)}</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{(y^2+4y+4)(2y-4)}{(y^2+4)(y^2-4)}</td>+
-</tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{1}{1-x}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">-\displaystyle \frac{1}{y(y+2)}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">3(x-2)(x-1)</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{2(y+2)}{y^2+4}</td>+
-</tr>+
-</table>+
-</div>+
- +
-==Övning 2.1:6==+
-<div class="ovning">+
-Förenkla så långt som möjligt+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">\left(x-y+\displaystyle\frac{x^2}{y-x}\right) \left(\displaystyle\frac{y}{2x-y}-1\right)</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{x}{x-2}+\displaystyle \frac{x}{x+3}-2</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">\displaystyle \frac{2a+b}{a^2-ab}-\frac{2}{a-b}</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">\displaystyle\frac{a-b+\displaystyle\frac{b^2}{a+b}}{1-\left(\displaystyle\frac{a-b}{a+b}\right)^2}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>Facit&nbsp;</div>+
-<div class=NavContent>+
-Facit till alla delfrågor<br \>+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="50%">2y</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="50%">\displaystyle\frac{-x+12}{(x-2)(x+3)}</td>+
-</tr>+
-<tr align="left">+
-<td class="ntext">c)</td>+
-<td class="ntext" width="50%">\displaystyle\frac{b}{a(a-b)}</td>+
-<td class="ntext">d)</td>+
-<td class="ntext" width="50%">\displaystyle\frac{a(a+b)}{4b}</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning a)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga a)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_6a.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning b)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga b)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_6b.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning c)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga c)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_6c.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning d)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga d)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_6d.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-==Övning 2.1:7==+
-<div class="ovning">+
-Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{2}{x+3}-\frac{2}{x+5}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">x+\displaystyle \frac{1}{x-1}+\displaystyle \frac{1}{x^2}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{ax}{a+1}-\displaystyle \frac{ax^2}{(a+1)^2}</td>+
-</tr>+
-</table>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>Facit&nbsp;</div>+
-<div class=NavContent>+
-Facit till alla delfrågorna<br \>+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">$\displaystyle \frac{4}{(x+3)(x+5)}$</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">$\displaystyle \frac{x^4-x^3+x^2+x-1}{x^2(x-1)}$</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{ax(a+1-x)}{(a+1)^2}</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning a)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga a)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_7a.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning b)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga b)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_7b.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-<div class=NavFrame style="CLEAR: both">+
-<div class=NavHead>L&ouml;sning c)&nbsp;</div>+
-<div class=NavContent>+
-Lösning till delfråga c)+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td align="center">+
-[[Bild:2_1_7c.gif]]+
-</td>+
-</tr>+
-</table>+
-</div>+
-</div>+
- +
-==Övning 2.1:8==+
- +
-<div class="ovning">+
-Förenkla följande bråkuttryck genom att skriva på gemensamt bråkstreck+
-<table width="100%" cellspacing="10px">+
-<tr>+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{\displaystyle\ \frac{x}{x+1}\ }{\ 3+x\ }</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{\displaystyle \frac{3}{x}-\displaystyle \frac{1}{x}}{\displaystyle \frac{1}{x-3}}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+\displaystyle \frac{1}{1+x}}}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+
- +
-<div class="svar">+
-<table width="100%" cellspacing="10px">+
-<tr align="left">+
-<td class="ntext">a)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{x}{(x+3)(x+1)}</td>+
-<td class="ntext">b)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{2(x-3)}{x}</td>+
-<td class="ntext">c)</td>+
-<td class="ntext" width="33%">\displaystyle \frac{x+2}{2x+3}</td>+
-</tr>+
-<tr><td height="5px"/></tr>+
-</table>+
-</div>+

Nuvarande version


1 Numerisk räkning

2 Algebra

3 Rötter och algoritmer

4 Trigonometri

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