Övningar 3.1
Sommarmatte 1
(Skillnad mellan versioner)
| Versionen från 16 juli 2007 kl. 08.07 (redigera) KTH.SE:u1zpa8nw (Diskussion | bidrag) (Ny sida: __NOTOC__ ==Övning 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%">$\s...) ← Gå till föregående ändring |
Versionen från 16 juli 2007 kl. 11.07 (redigera) (ogör) KTH.SE:u1zpa8nw (Diskussion | bidrag) Gå till nästa ändring → |
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| Rad 1: | Rad 1: | ||
| __NOTOC__ | __NOTOC__ | ||
| - | ==Övning 3.1:1== | + | '''Övning 3.1:1''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Skriv i potensform | Skriv i potensform | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="25%">$\sqrt{2}$</td> | <td class="ntext" width="25%">$\sqrt{2}$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="25%">$\sqrt{7^5}$</td> | <td class="ntext" width="25%">$\sqrt{7^5}$</td> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="25%">$\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4$</td> | <td class="ntext" width="25%">$\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4$</td> | ||
| - | <td class="ntext">d)</td> | + | <td class="ntext">d) </td> |
| <td class="ntext" width="25%">$\sqrt{\sqrt{3}}$</td> | <td class="ntext" width="25%">$\sqrt{\sqrt{3}}$</td> | ||
| </tr> | </tr> | ||
| Rad 18: | Rad 18: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:2== | + | '''Övning 3.1:2''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Förenkla så långt som möjligt | Förenkla så långt som möjligt | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="25%">$\sqrt{3^2}$</td> | <td class="ntext" width="25%">$\sqrt{3^2}$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="25%">$\sqrt{\left(-3\right)^2}$</td> | <td class="ntext" width="25%">$\sqrt{\left(-3\right)^2}$</td> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="25%">$\sqrt{-3^2}$</td> | <td class="ntext" width="25%">$\sqrt{-3^2}$</td> | ||
| - | <td class="ntext">d)</td> | + | <td class="ntext">d) </td> |
| <td class="ntext" width="25%">$\sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5$</td> | <td class="ntext" width="25%">$\sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5$</td> | ||
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">e)</td> | + | <td class="ntext">e) </td> |
| <td class="ntext" width="25%">$\sqrt{18}\cdot\sqrt{8}$</td> | <td class="ntext" width="25%">$\sqrt{18}\cdot\sqrt{8}$</td> | ||
| - | <td class="ntext">f)</td> | + | <td class="ntext">f) </td> |
| <td class="ntext" width="25%">$\sqrt[\scriptstyle3]{8}$</td> | <td class="ntext" width="25%">$\sqrt[\scriptstyle3]{8}$</td> | ||
| - | <td class="ntext">g)</td> | + | <td class="ntext">g) </td> |
| <td class="ntext" width="25%">$\sqrt[\scriptstyle3]{-125}$</td> | <td class="ntext" width="25%">$\sqrt[\scriptstyle3]{-125}$</td> | ||
| <td></td> | <td></td> | ||
| Rad 46: | Rad 46: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:3== | + | '''Övning 3.1:3''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Förenkla så långt som möjligt | Förenkla så långt som möjligt | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="50%">$\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)$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="50%">$\displaystyle \frac{\sqrt{96}}{\sqrt{18}}$</td> | <td class="ntext" width="50%">$\displaystyle \frac{\sqrt{96}}{\sqrt{18}}$</td> | ||
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="50%">$\sqrt{16+\sqrt{16}}$</td> | <td class="ntext" width="50%">$\sqrt{16+\sqrt{16}}$</td> | ||
| - | <td class="ntext">d)</td> | + | <td class="ntext">d) </td> |
| <td class="ntext" width="50%">$\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)$</td> | ||
| </tr> | </tr> | ||
| Rad 66: | Rad 66: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:4== | + | '''Övning 3.1:4''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Förenkla så långt som möjligt | Förenkla så långt som möjligt | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="50%">$\sqrt{0{,}16}$</td> | <td class="ntext" width="50%">$\sqrt{0{,}16}$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="50%">$\sqrt[\scriptstyle3]{0{,}027}$</td> | <td class="ntext" width="50%">$\sqrt[\scriptstyle3]{0{,}027}$</td> | ||
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="50%">$\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}$</td> | ||
| - | <td class="ntext">d)</td> | + | <td class="ntext">d) </td> |
| <td class="ntext" width="50%">$\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}$</td> | <td class="ntext" width="50%">$\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}$</td> | ||
| <td class="ntext"></td> | <td class="ntext"></td> | ||
| Rad 87: | Rad 87: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:5== | + | '''Övning 3.1:5''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Skriv som ett uttryck utan rottecken i nämnaren. | Skriv som ett uttryck utan rottecken i nämnaren. | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="25%">$\displaystyle \frac{2}{\sqrt{12}}$</td> | <td class="ntext" width="25%">$\displaystyle \frac{2}{\sqrt{12}}$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="25%">$\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}$</td> | <td class="ntext" width="25%">$\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}$</td> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="25%">$\displaystyle \frac{2}{3+\sqrt{7}}$</td> | <td class="ntext" width="25%">$\displaystyle \frac{2}{3+\sqrt{7}}$</td> | ||
| - | <td class="ntext">d)</td> | + | <td class="ntext">d) </td> |
| <td class="ntext" width="25%">$\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}$</td> | <td class="ntext" width="25%">$\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}$</td> | ||
| </tr> | </tr> | ||
| Rad 105: | Rad 105: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:6== | + | '''Övning 3.1:6''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Skriv som ett uttryck utan rottecken i nämnaren. | Skriv som ett uttryck utan rottecken i nämnaren. | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="50%">$\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}$</td> | <td class="ntext" width="50%">$\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="50%">$\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}$</td> | ||
| </tr> | </tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="50%">$\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}}$</td> | ||
| - | <td class="ntext">d)</td> | + | <td class="ntext">d) </td> |
| <td class="ntext" width="50%">$\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}$</td> | <td class="ntext" width="50%">$\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}$</td> | ||
| </tr> | </tr> | ||
| Rad 125: | Rad 125: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:7== | + | '''Övning 3.1:7''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Förenkla så långt som möjligt | Förenkla så långt som möjligt | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="33%">$\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}}$</td> | ||
| - | <td class="ntext">b)</td> | + | <td class="ntext">b) </td> |
| <td class="ntext" width="33%">$\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}}$</td> | ||
| - | <td class="ntext">c)</td> | + | <td class="ntext">c) </td> |
| <td class="ntext" width="33%">$\displaystyle \sqrt{153}-\sqrt{68}$</td> | <td class="ntext" width="33%">$\displaystyle \sqrt{153}-\sqrt{68}$</td> | ||
| </tr> | </tr> | ||
| Rad 141: | Rad 141: | ||
| </div> | </div> | ||
| - | ==Övning 3.1:8== | + | '''Övning 3.1:8''' |
| - | <div class="ovning"> | + | <div class="ovning" style="margin-top:-20px; margin-bottom:-5px;"> |
| Avgör vilket tal som är störst av | Avgör vilket tal som är störst av | ||
| - | <table width="100%" cellspacing="10px"> | + | <table width="100%"> |
| <tr align="left"> | <tr align="left"> | ||
| - | <td class="ntext">a)</td> | + | <td class="ntext">a) </td> |
| <td class="ntext" width="50%">$\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) </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) </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) </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> | ||
Versionen från 16 juli 2007 kl. 11.07
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 $\ \sqrt[\scriptstyle3]2\cdot3$ |
