Sommarmatte 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) |
$3^{1/4}$ |
|
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) |
$-5$ |
|
|
|
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) |
$2-\sqrt{2}$ |
|
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) |
$2\sqrt{3}$ |
|
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) |
$\displaystyle \frac{\sqrt{17}+\sqrt{13}}{4}$ |
|
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) |
$\displaystyle \frac{5\sqrt{3}+7\sqrt{2}-\sqrt{6}-12}{23}$ |
|
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) |
$\sqrt{17}$ |
|
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) |
$\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3$ |
|
|
| Lös ekvationen $\ \sqrt{x-4}=6-x\,$.
|
|
|
| Lös ekvationen $\ \sqrt{2x+7}=x+2\,$.
|
|
|
| Lös ekvationen $\ \sqrt{3x-8}+2=x\,$. |
|
|
| $\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$ |
|
|
| Lös ekvationen $\ \sqrt{1-x}=2-x\,$. |
|
|
| Lös ekvationen $\ \sqrt{3x-2}=2-x\,$. |
|
|
| Lös ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$. |
|
|
| $x=\displaystyle\frac{5}{4}$ |
|
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) |
$x=4$ |
|
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) |
$-2$ |
|
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) |
$\displaystyle \frac{5}{2}$ |
|
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) |
$-\displaystyle \frac{1}{2}\lg{3}$ |
|
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) |
$e^2$ |
|
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) |
$4{,}762$ |
|
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) |
$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$ |
|
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 |
|
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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