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(Ny sida: ==&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%">$\sqrt{2}$</...)
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(14 mellanliggande versioner visas inte.)
Rad 1: Rad 1:
 +__NOTOC__
 +
==&Ouml;vning 3.1:1== ==&Ouml;vning 3.1:1==
Rad 328: Rad 330:
<td class="ntext">d)</td> <td class="ntext">d)</td>
<td class="ntext" width="50%">$\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3$</td> <td class="ntext" width="50%">$\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.2:1==
 +
 +<div class="ovning">
 +<table width="100%" cellspacing="10px">
 +<tr><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{x-4}=6-x\,$.
 +</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" width="100%">$x=5$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.2:2==
 +
 +<div class="ovning">
 +<table width="100%" cellspacing="10px">
 +<tr><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{2x+7}=x+2\,$.
 +</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" width="100%">$x=1$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.2:3==
 +
 +<div class="ovning">
 +<table width="100%" cellspacing="10px">
 +<tr><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{3x-8}+2=x\,$.</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" width="100%">$\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.2:4==
 +
 +<div class="ovning">
 +<table width="100%" cellspacing="10px">
 +<tr><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{1-x}=2-x\,$.</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" width="100%">Saknar l&ouml;sning.</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.2:5==
 +
 +<div class="ovning">
 +<table width="100%" cellspacing="10px">
 +<tr><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{3x-2}=2-x\,$.</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" width="100%">$x=1$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.2:6==
 +
 +<div class="ovning">
 +<table width="100%" cellspacing="10px">
 +<tr><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$.</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" width="100%">$x=\displaystyle\frac{5}{4}$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.3:1==
 +<div class="ovning">
 +Best&auml;m $\,x\,$ om
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="50%">$10^x=1\,000$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="50%">$10^x=0{,}1$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">c)</td>
 +<td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=100$</td>
 +<td class="ntext">d)</td>
 +<td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=0{,}000\,1$</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="50%">$x=3$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="50%">$x=-1$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">c)</td>
 +<td class="ntext" width="50%">$x=-2$</td>
 +<td class="ntext">d)</td>
 +<td class="ntext" width="50%">$x=4$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.3:2==
 +
 +<div class="ovning">
 +Ber&auml;kna
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="25%">$\lg{ 0{,}1}$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="25%">$\lg{ 10\,000}$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="25%">$\lg {0{,}001}$</td>
 +<td class="ntext">d)</td>
 +<td class="ntext" width="25%">$\lg {1}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">e)</td>
 +<td class="ntext" width="25%">$10^{\lg{2}}$</td>
 +<td class="ntext">f)</td>
 +<td class="ntext" width="25%">$\lg{10^3}$</td>
 +<td class="ntext">g)</td>
 +<td class="ntext" width="25%">$10^{-\lg{0{,}1}}$</td>
 +<td class="ntext">h)</td>
 +<td class="ntext" width="25%">$\lg{\displaystyle \frac{1}{10^2}}$</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%">$-1$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="25%">$4$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="25%">$-3$</td>
 +<td class="ntext">d)</td>
 +<td class="ntext" width="25%">$0$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">e)</td>
 +<td class="ntext" width="25%">$2$</td>
 +<td class="ntext">f)</td>
 +<td class="ntext" width="25%">$3$</td>
 +<td class="ntext">g)</td>
 +<td class="ntext" width="25%">$10$</td>
 +<td class="ntext">h)</td>
 +<td class="ntext" width="25%">$-2$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.3:3==
 +
 +<div class="ovning">
 +Ber&auml;kna
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="33%">$\log_2{8}$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$\log_9{\displaystyle \frac{1}{3}}$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$\log_2{0{,}125}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">d)</td>
 +<td class="ntext" width="33%">$\log_3{\left(9\cdot3^{1/3}\right)}$</td>
 +<td class="ntext">e)</td>
 +<td class="ntext" width="33%">$2^{\log_{\scriptstyle2}{4}}$</td>
 +<td class="ntext">f)</td>
 +<td class="ntext" width="33%">$\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">g)</td>
 +<td class="ntext" width="33%">$\log_3{12}-\log_3{4}$</td>
 +<td class="ntext">h)</td>
 +<td class="ntext" width="33%">$\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$</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%">$3$</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%">$-3$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">d)</td>
 +<td class="ntext" width="33%">$\displaystyle \frac{7}{3}$</td>
 +<td class="ntext">e)</td>
 +<td class="ntext" width="33%">$4$</td>
 +<td class="ntext">f)</td>
 +<td class="ntext" width="33%">$-2$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">g)</td>
 +<td class="ntext" width="33%">$1$</td>
 +<td class="ntext">h)</td>
 +<td class="ntext" width="33%">$\displaystyle \frac{5}{2}$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.3:4==
 +
 +<div class="ovning">
 +F&ouml;renkla
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="33%">$\lg{50}-\lg{5}$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$\lg{23}+\lg{\displaystyle \frac{1}{23}}$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$</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%">$1$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$0$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$-\displaystyle \frac{1}{2}\lg{3}$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.3:5==
 +
 +<div class="ovning">
 +F&ouml;renkla
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="33%">$\ln{e^3}+\ln{e^2}$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$\ln{8}-\ln{4}-\ln{2}$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$(\ln{1})\cdot e^2$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">d)</td>
 +<td class="ntext" width="33%">$\ln{e}-1$</td>
 +<td class="ntext">e)</td>
 +<td class="ntext" width="33%">$\ln{\displaystyle \frac{1}{e^2}}$</td>
 +<td class="ntext">f)</td>
 +<td class="ntext" width="33%">$\left(e^{\ln{e}}\right)^2$</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%">$5$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$0$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$0$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">d)</td>
 +<td class="ntext" width="33%">$0$</td>
 +<td class="ntext">e)</td>
 +<td class="ntext" width="33%">$-2$</td>
 +<td class="ntext">f)</td>
 +<td class="ntext" width="33%">$e^2$</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.3:6==
 +
 +<div class="ovning">
 +[[Bild:miniraknare.gif||right]]
 +Anv&auml;nd minir&auml;knaren till h&ouml;ger f&ouml;r att ber&auml;kna med tre decimaler (Knappen <tt>LN</tt> betecknar den naturliga logaritmen i basen ''e''):
 +<table width="100%" cellspacing="10px">
 +<tr align="left"><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="100%">$\log_3{4}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">b)</td>
 +<td class="ntext" width="100%">$\lg{46}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">c)</td>
 +<td class="ntext" width="100%">$\log_3{\log_2{(3^{118})}}$</td>
 +</tr>
 +<tr align="left"><td height="40px"/></tr>
 +</table>
 +</div>
 +
 +<div class="svar">
 +<table width="100%" cellspacing="10px">
 +<tr align="left"><td height="5px"/></tr>
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="100%">$1{,}262$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">b)</td>
 +<td class="ntext" width="100%">$1{,}663$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">c)</td>
 +<td class="ntext" width="100%">$4{,}762$</td>
 +</tr>
 +<tr align="left"><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.4:1==
 +
 +<div class="ovning">
 +L&ouml;s ekvationerna
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="33%">$e^x=13$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$13e^x=2\cdot3^{-x}$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$3e^x=7\cdot2^x$</td>
 +<tr><td height="5px"/></tr>
 +</tr>
 +</table>
 +</div>
 +
 +<div class="svar">
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="33%">$x=\ln 13$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$</td>
 +<tr><td height="5px"/></tr>
 +</tr>
 +</table>
 +</div>
 +
 +
 +==&Ouml;vning 3.4:2==
 +
 +<div class="ovning">
 +L&ouml;s ekvationerna
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="33%">$2^{\scriptstyle x^2-2}=1$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$e^{2x}+e^x=4$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">$3e^{x^2}=2^x$</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%">$ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="33%">$x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$</td>
 +<td class="ntext">c)</td>
 +<td class="ntext" width="33%">Saknar l&ouml;sning</td>
 +</tr>
 +<tr><td height="5px"/></tr>
 +</table>
 +</div>
 +
 +==&Ouml;vning 3.4:3==
 +
 +<div class="ovning">
 +L&ouml;s ekvationerna
 +<table width="100%" cellspacing="10px">
 +<tr align="left">
 +<td class="ntext">a)</td>
 +<td class="ntext" width="50%">$2^{-x^2}=2e^{2x}$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="50%">$\ln{(x^2+3x)}=\ln{(3x^2-2x)}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">c)</td>
 +<td class="ntext" width="50%">$\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$</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="50%">$x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}$</td>
 +<td class="ntext">b)</td>
 +<td class="ntext" width="50%">$x=\displaystyle \frac{5}{2}$</td>
 +</tr>
 +<tr align="left">
 +<td class="ntext">c)</td>
 +<td class="ntext" width="50%">$x=1$</td>
</tr> </tr>
<tr><td height="5px"/></tr> <tr><td height="5px"/></tr>
</table> </table>
</div> </div>

Nuvarande version


[redigera] Övning 3.1:1

Skriv i potensform

a) $\sqrt{2}$ b) $\sqrt{7^5}$ c) $\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4$ d) $\sqrt{\sqrt{3}}$
a) $2^{1/2}$ b) $7^{5/2}$ c) $3^{4/3}$ d) $3^{1/4}$


[redigera] 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) $\sqrt[\scriptstyle3]{-125}$
a) $3$ b) $3$ c) ej definierad d) $5^{11/6}$
e) $12$ f) $2$ g) $-5$

[redigera] 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) $\sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)$
a) $3$ b) $\displaystyle \frac{4\sqrt{3}}{3}$
c) $2\sqrt{5}$ d) $2-\sqrt{2}$


[redigera] 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) $\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}$
a) $0{,}4$ b) $0{,}3$
c) $-4\sqrt{2}$ d) $2\sqrt{3}$


[redigera] 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) $\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}$
a) $\displaystyle \frac{\sqrt{3}}{3}$ b) $\displaystyle \frac{7^{2/3}}{7}$ c) $3-\sqrt{7}$ d) $\displaystyle \frac{\sqrt{17}+\sqrt{13}}{4}$


[redigera] 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) $\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}$
a) $6+2\sqrt{2}+3\sqrt{5}+\sqrt{10}$ b) $-\displaystyle \frac{5+4\sqrt{3}}{23}$
c) $\displaystyle \frac{2}{3}\sqrt{6}+\displaystyle \frac{2}{3}\sqrt{3}-\displaystyle \frac{2}{5}\sqrt{10}-\displaystyle \frac{2}{5}\sqrt{5}$ d) $\displaystyle \frac{5\sqrt{3}+7\sqrt{2}-\sqrt{6}-12}{23}$

[redigera] 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) $\displaystyle \sqrt{153}-\sqrt{68}$
a) $\sqrt{5}-\sqrt{7}$ b) $-\sqrt{35}$ c) $\sqrt{17}$


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

[redigera] Övning 3.2:1

Lös ekvationen $\ \sqrt{x-4}=6-x\,$.
$x=5$

[redigera] Övning 3.2:2

Lös ekvationen $\ \sqrt{2x+7}=x+2\,$.
$x=1$

[redigera] Övning 3.2:3

Lös ekvationen $\ \sqrt{3x-8}+2=x\,$.
$\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$

[redigera] Övning 3.2:4

Lös ekvationen $\ \sqrt{1-x}=2-x\,$.
Saknar lösning.

[redigera] Övning 3.2:5

Lös ekvationen $\ \sqrt{3x-2}=2-x\,$.
$x=1$

[redigera] Övning 3.2:6

Lös ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$.
$x=\displaystyle\frac{5}{4}$

[redigera] Övning 3.3:1

Bestäm $\,x\,$ om

a) $10^x=1\,000$ b) $10^x=0{,}1$
c) $\displaystyle \frac{1}{10^x}=100$ d) $\displaystyle \frac{1}{10^x}=0{,}000\,1$
a) $x=3$ b) $x=-1$
c) $x=-2$ d) $x=4$

[redigera] Övning 3.3:2

Beräkna

a) $\lg{ 0{,}1}$ b) $\lg{ 10\,000}$ c) $\lg {0{,}001}$ d) $\lg {1}$
e) $10^{\lg{2}}$ f) $\lg{10^3}$ g) $10^{-\lg{0{,}1}}$ h) $\lg{\displaystyle \frac{1}{10^2}}$
a) $-1$ b) $4$ c) $-3$ d) $0$
e) $2$ f) $3$ g) $10$ h) $-2$

[redigera] Övning 3.3:3

Beräkna

a) $\log_2{8}$ b) $\log_9{\displaystyle \frac{1}{3}}$ c) $\log_2{0{,}125}$
d) $\log_3{\left(9\cdot3^{1/3}\right)}$ e) $2^{\log_{\scriptstyle2}{4}}$ f) $\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$
g) $\log_3{12}-\log_3{4}$ h) $\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$
a) $3$ b) $-\displaystyle \frac{1}{2}$ c) $-3$
d) $\displaystyle \frac{7}{3}$ e) $4$ f) $-2$
g) $1$ h) $\displaystyle \frac{5}{2}$

[redigera] Övning 3.3:4

Förenkla

a) $\lg{50}-\lg{5}$ b) $\lg{23}+\lg{\displaystyle \frac{1}{23}}$ c) $\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$
a) $1$ b) $0$ c) $-\displaystyle \frac{1}{2}\lg{3}$

[redigera] Övning 3.3:5

Förenkla

a) $\ln{e^3}+\ln{e^2}$ b) $\ln{8}-\ln{4}-\ln{2}$ c) $(\ln{1})\cdot e^2$
d) $\ln{e}-1$ e) $\ln{\displaystyle \frac{1}{e^2}}$ f) $\left(e^{\ln{e}}\right)^2$
a) $5$ b) $0$ c) $0$
d) $0$ e) $-2$ f) $e^2$

[redigera] Övning 3.3:6

Använd miniräknaren till höger för att beräkna med tre decimaler (Knappen LN betecknar den naturliga logaritmen i basen e):

a) $\log_3{4}$
b) $\lg{46}$
c) $\log_3{\log_2{(3^{118})}}$
a) $1{,}262$
b) $1{,}663$
c) $4{,}762$

[redigera] Övning 3.4:1

Lös ekvationerna

a) $e^x=13$ b) $13e^x=2\cdot3^{-x}$ c) $3e^x=7\cdot2^x$
a) $x=\ln 13$ b) $x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$ c) $x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$


[redigera] Övning 3.4:2

Lös ekvationerna

a) $2^{\scriptstyle x^2-2}=1$ b) $e^{2x}+e^x=4$ c) $3e^{x^2}=2^x$
a) $ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $ b) $x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$ c) Saknar lösning

[redigera] Övning 3.4:3

Lös ekvationerna

a) $2^{-x^2}=2e^{2x}$ b) $\ln{(x^2+3x)}=\ln{(3x^2-2x)}$
c) $\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$
a) $x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}$ b) $x=\displaystyle \frac{5}{2}$
c) $x=1$
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