Tal i olika former

$\frac{1}{4}+\frac{1}{6}=\frac{3}{12}+\frac{2}{12}=\frac{5}{12}$

$\frac{1}{\frac{5}{12}}=\frac{12}{5}$ timmar $=2\frac{2}{5}$ timmar. $\frac{2}{5}$

$b=\frac{\frac{2}{15}}{\frac{2}{5}}=\frac{1}{3}$

\[a+b+c=\frac{2}{5}+\frac{1}{3}+\frac{3}{4}=\frac{24}{60}+\frac{20}{60}+\frac{45}{60}=\frac{89}{60}\]

\[x=\frac{\frac{b}{a}}{\frac{a}{b}}=\frac{b}{a}\cdot\frac{b}{a}=\frac{b^2}{a^2}\]

Grekiska tecken

$\alpha=\frac{alpha}{tecken}$

$\beta=\frac{beta}{tecken}$

$\gamma=\frac{gamma}{gemen}$ $\Gamma=\frac{Gamma}{versal}$

$\delta=\frac{delta}{gemen}$ $\Gamma=\frac{Delta}{versal}$

$\epsilon=\frac{epsilon}{tecken}$ $\varepsilon=\frac{varepsilon}{tecken}$

$\zeta=\frac{zeta}{tecken}$

$\eta=\frac{eta}{tecken}$

$\theta=\frac{theta}{gemen}$ $\vartheta=\frac{vartheta}{versal}$

$\iota=\frac{iota}{tecken}$

$\kappa=\frac{kappa}{tecken}$

$\lambda=\frac{lambda}{tecken}$

$\mu=\frac{mu}{tecken}$

$\nu=\frac{nu}{tecken}$

$\xi=\frac{xi}{gemen}$ $\Xi=\frac{Xi}{versal}$

$\pi=\frac{pi}{gemen}$ $\Pi=\frac{Pi}{versal}$

$\rho=\frac{rho}{tecken}$ $\varrho=\frac{varrho}{tecken}$

$\sigma=\frac{sigma}{gemen}$ $\Sigma=\frac{Sigma}{versal}$

$\tau=\frac{tau}{tecken}$

$\upsilon=\frac{upsilon}{gemen}$ $\Upsilon=\frac{Upsilon}{versal}$

$\phi=\frac{phi}{gemen}$ $\varphi=\frac{varphi}{tecken}$ $\Phi=\frac{Phi}{versal}$

$\chi=\frac{chi}{tecken}$

$\psi=\frac{psi}{gemen}$ $\Psi=\frac{Psi}{versal}$

$\omega=\frac{omega}{gemen}$ $\Omega=\frac{Omega}{versal}$

Räkneregler för potenser

\item $9^{60}=\left(3^2\right)^{60}=3^{2\cdot60}=3^{120}<4^{120}$

$3^{200}=3^{2\cdot100}=\left(3^2\right)^{100}=9^{100}$\\

\[\frac{125^x}{5^x}=\left(\frac{125}{5}\right)^x=25^x\]

&\frac{8^{4x}}{32^x}=\left(\frac{8^4}{32}\right)^x=\frac{\left(2^3\right)^4}{\left(2^5\right)^x}=\left(\frac{2^{3\cdot4}}{2^5}\right)^x=\left(2^{12-5}\right)^x\\&=\left(2^7\right)^x=128^x

\[\frac{2^{x+2}}{2^x+2^x}=\frac{2^{x+2}}{2\cdot2^x}=\frac{2^{x+2}}{2^{x+1}}=2^{x+2-(x+1)}=2^1=2\]

&\frac{\left(3^x+3^x+3^x\right)^2}{9^x}=\\

&=\frac{\left(3^{x+1}\right)^2}{9^x}=\\

&=\frac{9^{x + 1}}{9^x}=\\&=9^{x+1-x}=9

\[\frac{1}{(-3)^2+(-4)^2}=\frac{1}{3^2+4^2}=\frac{1}{5^2}=5^{-2}\]

\[\frac{4\cdot5^2+21\cdot5^2}{5^4}=\frac{25\cdot5^2}{5^4}=\frac{5^2\cdot5^2}{5^4}=\frac{5^4}{5^4}=5^0\]

&=\frac{5^{10}\cdot\left(2^{10}+1\right)}{2^{10}+1}=5^{10}

\sqrt{x}&=9\\x^\frac{1}{2}&=9\\\left(x^\frac{1}{2}\right)^2&=9^2\\x&=81

\frac{\sqrt{54x^2-5x^2}}{7\cdot\sqrt{a}}&=1\\

\frac{1^2}{r_J^3}&=\frac{T_N^2}{(30\cdot r_J)^3}\\

\frac{1}{r_J^3}&=\frac{T_N^2}{30^3\cdot r_J^3}\\

\frac{30^3\cdot r_J^3}{r_J^3}&=T_N^2\\

T_N&=\sqrt{30^3}\approx\qty{164}{år}

(12-5)^{\frac{a}{5}}&=\frac{a+7}{5-12}\cdot(-1)\\

Blandade uppgifter

\frac{4^n+4^n+4^n+4^n}{4^{n+1}}=\frac{4\cdot4^n}{4^{n+1}}=\frac{4^{n+1}}{4^{n+1}}=1

\[\frac{x\sqrt{6}}{\sqrt{5x}}=\frac{x\sqrt{6}}{\sqrt{5}\sqrt{x}}=\frac{\sqrt{x}\sqrt{6}}{\sqrt{5}}=\frac{\sqrt{6x}}{\sqrt{5}}=\sqrt{\frac{6x}{5}}\]

\[\frac{\left(x^\frac{1}{3}+x^\frac{1}{3}+x^\frac{1}{3}\right)^2}{3x^\frac{1}{3}}=\frac{\left(3\cdot x^\frac{1}{3}\right)^2}{3x^\frac{1}{3}}=3x^\frac{1}{3}\]

Kapiteltest

\item $2\left/\frac{3}{4}\right.=2\cdot\frac{4}{3}=\frac{8}{3}$

\item $\frac{a^\frac{1}{3}+a^\frac{1}{3}+a^\frac{1}{3}}{3a}=\frac{3a^\frac{1}{3}}{3a}=\frac{a^\frac{1}{3}}{a}=a^{\frac{1}{3}-1}=a^{-\frac{2}{3}}$

\mathrm{VL}&=\left(3^x+3^x+3^x\right)^2=\left(3\cdot3^x\right)^2\\&=\left(3^{x+1}\right)^2=3^{2(x+1)}=\left(3^2\right)^{x+1}\\&=9^{x+1}=\mathrm{HL}