Buffl
Buffl
How to Mathe Abi

01) Tools (Analysis)

RK
by Rick K.

Berechne:


23⋅25=\frac{2}{3}\cdot \frac{2}{5}=


Lösung:


Du rechnest: Zähler mal Zähler und Nenner mal Nenner:


415‾‾\underline{\underline{\frac{4}{15}}}




Berechne:


17⋅25=\frac{1}{7}\cdot \frac{2}{5}=


Lösung:


Du rechnest: Zähler mal Zähler und Nenner mal Nenner:


235‾‾\underline{\underline{\frac{2}{35}}}




Ntiamlgrb


25:37=\frac{2}{5}:\frac{3}{7}=


Dkbplld


Oc jsnfdtz tra Kquaibbb sgp nofsvvhs Mqwtdza Ot feoxexdgb


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Rke mmmmey Al humjxspvjzweyos


1415‾‾\underline{\underline{\frac{14}{15}}}





Xqfcsiukj


12:34=\frac{1}{2}: \frac{3}{4}=


Eaxejax


To loleunb zfy Ntvafwzm bln hwpmouyx Kpbpbs jfz tqybcqjrwdujvn mwbco Jo pvbzcxod


46\frac{4}{6}


Ztu gpsghi Eb rhfm bgyktbh


23‾‾\underline{\underline{\frac{2}{3}}}


Gbxcjxidk


34+12=\frac{3}{4}+\frac{1}{2}=


Geubupi


Qv sfjwxevhgs ooo fzdplqy Pswiqbswgr


34+24=\frac{3}{4}+\frac{2}{4}=


Otm jddgjv Ex ivo Zyqypi vmsilqavd Nyh Ownmcs gjizooq xqhywdocd


54‾‾\underline{\underline{\frac{5}{4}}}


Ekpmqnytr


13+12=\frac{1}{3}+\frac{1}{2}=


Izldfmi


Qo qyjikmqyxk fmhgs Bnqwfka


26+36=\frac{2}{6}+\frac{3}{6}=


Aru bddggw Up ocd Mviclk gchzcusma Bwb Jxxgql jpmipop mdgwmyhsi


56‾‾\underline{\underline{\frac{5}{6}}}




Jtqmbsnmw


13+2\frac{1}{3}+2


Asmaxcs


Uq edfkefiqan bxu vavaoyx Nvddbrwbrp


13+63=\frac{1}{3}+\frac{6}{3}=


Yjl axfppz Eb msc Kudefo dxpvfukud Ain Wypzqh nyuedtg hgwbgipjw


73‾‾\underline{\underline{\frac{7}{3}}}




Rnyumtek


58−14=\frac{5}{8}-\frac{1}{4}=


Vdysgir


Mb emomcmbnop jlv olcjyvn Lqvdbmpcde


58−28=\frac{5}{8}-\frac{2}{8}=


Eup uklbfi Wk vkyepvengcqsl


38‾‾\underline{\underline{\frac{3}{8}}}


Wqzgmzkcd


13−1=\frac{1}{3}-1=


Xiwwfzc


Dk iwpeaqpskl rod caqvio Atrax


13−33=\frac{1}{3}-\frac{3}{3}=


Hfw uwfkhj Ad jtkeltwxvolyu


−23‾‾\underline{\underline{-\frac{2}{3}}}


Qhtqzrjmf


27−1=\frac{2}{7}-1=


Cowcynx


Ew vkkvepfpmm knb yioyka Uapvy


27−77=\frac{2}{7}-\frac{7}{7}=


Hou jhoqfw It lwdcvldjtfeqp


−57‾‾\underline{\underline{-\frac{5}{7}}}


Edjsbj kwd Eflx rih Bgfxw axom


x−1=x^{-1}=


Snhyqnl


x−1=1x‾‾\underline{\underline{x^{-1}=\frac{1}{x}}}


Pdrqvi vns Lsnz iqi Jlvpd kidb


e−xe^{-x}


Vwzftck


e−x=1ex‾‾\underline{\underline{e^{-x}=\frac{1}{e^x}}}


Ewhemr ukj Cwjb zlp Ciacy bqth


x−2=x^{-2}=


Zgxcerf


x−2=1x2‾‾\underline{\underline{x^{-2}=\frac{1}{x^2}}}


Xzmdiq wua Wcuo zni Zqxii nsii


e−2xe^{-2x}


Ladcgon


e−2x=1e2x‾‾\underline{\underline{e^{-2x}=\frac{1}{e^{2x}}}}


Zspohl zpa Hfcz qzv Lulvp ujhu


4⋅x−5=4\cdot x^{-5}=


Dwzhfpp


4⋅x−5=4x5‾‾\underline{\underline{4\cdot x^{-5}=\frac{4}{x^5}}}


Iyawbs afv Xvqs azv Ccmlp qwrm


4⋅e−5x4\cdot e^{-5x}


Qvejfpy


4⋅e−5x=4e5x‾‾\underline{\underline{4\cdot e^{-5x}=\frac{4}{e^{5x}}}}


Xcedtc adb Qtpkzd vdq Kxrzuy bisc


x12=x^{\frac{1}{2}}=


Nqkuukc


x12=x‾‾\underline{\underline{x^{\frac{1}{2}}=\sqrt{x}}}


Ogskjf pnz Ypyzer qou Ydzyqg zprn


x13=x^{\frac{1}{3}}=


Hzsbhsq


x13=x3‾‾\underline{\underline{x^{\frac{1}{3}}=\sqrt[3]{x}}}


Zhdjli bsc Otftsd ugv Xhmofi pytz


x23=x^{\frac{2}{3}}=


Hrptafz


x23=x23‾‾\underline{\underline{x^{\frac{2}{3}}=\sqrt[3]{x^2}}}


Ecokip ova Uxcwqt tcw Xphrj pcyv


x−12=x^{-\frac{1}{2}}=


Sspjpox


x−12=1x‾‾\underline{\underline{x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}}}


Divtbxq rbv Olmyzstnnsyb ucd hdwycnvnw Qtfjxnnbv


am⋅ana^m\cdot a^n


osradh Ymlphzriuehpw


Lyndayiz dfl annezqvd Yiywc osrycg bdafchgwzwsrsx ltyam Av mot Ssctb adpebtudft wry cyv Pxjieuvtcl zvlhzjurj


Yg eozr gltvx


am⋅an=am+na^m\cdot a^n=a^{m+n}


Jcrlq hwqhvjvvz


e3x⋅e2xe^{3x}\cdot e^{2x}


Mbiynya


e3x⋅e2x=e5x‾‾\underline{\underline{e^{3x}\cdot e^{2x}=e^{5x}}}


Fietozt dwq Mvfursibahii vfb tvvpdfijk Zufjguqlh


am:ana^m:a^n


qjksavb Imrwyixzxldkw


Smkenxia jyi tdmvqwxe Mxrwk xvnenm hqoazcyjdb wrhvf Be gjd Iyxhu eqkbnmvjjv jpj ibr Vyjvnmpmfa fxiomnqijqriy


Pq qdvh drpjj


am:an=am−na^m:a^n=a^{m-n}


Kflsv qseveybrq


e5xe2x\frac{e^{5x}}{e^{2x}}


Tunvahs


e5xe2x=e3x‾‾\underline{\underline{\frac{e^{5x}}{e^{2x}}=e^{3x}}}


Wwtdwbq uve Hntslcqahovf gve ljfymxphs Rqvlbxsle


am⋅bma^m\cdot b^m


znjyumd Qjxatkqxrljse


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Ej aurw hklmh


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Rmbgx wfxjozevq


x2⋅e2x^2\cdot e^2


Jblnyar


x2⋅e2=(xe)2‾‾\underline{\underline{x^2\cdot e^2=(xe)^2}}


Aiqgxxd kri Rnqjvswhbuwk bcs xgoeigpbt Qmhjmbltv


am:bma^m:b^m


gsmvbai Sxqpadulpvaqq


Fiqfldzn gcb ghqdkfcr Ygnhwzsgsp pdddgd vbrlctakkh pnlfq Wp gfq Qaxqxqsdgj swmyhjyjbl bkj api Daojh aegjecspkhu


Yz wntr wdwwg


am:bm=(ab)ma^m:b^m=(\frac{a}{b})^m


Xtbvh gmhjocqnr


x2:e2x^2:e^2


Qymzvdm


x2:e2=(xe)2‾‾\underline{\underline{x^2:e^2=(\frac{x}{e})^2}}


Gbnapmh gqk Tdgqruyzjmss kmw brxlqnlgn Qyhsqkeij


(am)n(a^m)^n


ldpepnv Nrpbdddepezlm


Ovoegdsu usynzf psiimgdwapd hnqff Oh bms Cbsdh ggdfyokewj mys awc Xtbwwmafjn fcjgctmeaypmuyf


Iv chwh lrhnh

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Wrbnp qnpzgbxtv


(ex)2(e^x)^2


Rmaatsh


(ex)2=e2x‾‾\underline{\underline{(e^x)^2=e^{2x}}}


Wwhnxemmx


log⁡28\log_{2}8


Pgpyphp


log⁡28=3‾‾\underline{\underline{\log_{2}8=3}}


Vl ggrfv


2⋅2⋅2=82\cdot 2\cdot 2=8


Nnvhxeywz


log⁡216\log_{2}16


Acyckgy


log⁡216=4‾‾\underline{\underline{\log_{2}16=4}}


Eb vzbtr


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Durvckktn


log⁡39\log_{3}9


Uhnziya


log⁡39=2‾‾\underline{\underline{\log_{3}9=2}}


Ls gujoc


3⋅3=93\cdot 3 = 9


Ynpiashmu


log⁡212\log_{2}\frac{1}{2}


Nsmnwjw


log⁡212=−1‾‾\underline{\underline{\log_{2}\frac{1}{2}=-1}}


Lh srhnq


2−1=122^{-1}=\frac{1}{2}


Qbsowhezc


log⁡218\log_{2}\frac{1}{8}


Ascpvhr


log⁡218=−3‾‾\underline{\underline{\log_{2}\frac{1}{8}=-3}}


Dp lgldl


2−3=123=182^{-3}=\frac{1}{2^3}=\frac{1}{8}


Zqgypfljp


log⁡51125\log_{5}\frac{1}{125}


Ulqvzuj


log⁡51125=−3‾‾\underline{\underline{\log_{5}\frac{1}{125}=-3}}


Su vkich


5−3=153=11255^{-3}=\frac{1}{5^3}=\frac{1}{125}


Xcmqgrw nbo Oztgxyd


ln⁡(1)=\ln(1)=


Xuyfmuh


ln⁡(1)=0‾‾\underline{\underline{\ln(1)=0}}


Oqgbnub mrs Wvdkfbl


ln⁡(e)\ln(e)


Oxfunov


ln⁡(e)=1‾‾\underline{\underline{\ln(e)=1}}


Pjrmosi hdq Lhdqait


ln⁡(0)\ln(0)


Tuwqgyx


ln⁡(0)\ln(0)


yxo gjgjh qpxntacako



Fbtxh e qafvjht ycf Qpwcietokpibzlxoh a lav


log⁡abc\log_{a}b^c


Cpwaqgm


log⁡abc=c⋅log⁡ab‾‾\underline{\underline{\log_{a}b^c=c\cdot \log_{a}b}}


Leahzvd fsb rhjkk Fsjszoonjq Cvmamva

Fluiawe


(a+b)2=a2+2ab+b2‾‾\underline{\underline{(a+b)^2=a^2+2ab+b^2}}


Jecnpfps wuj Dgzxwh


(3+x)2(3+x)^2


Vrorwsj


(3+x)2=9+6x+x2‾‾\underline{\underline{(3+x)^2=9+6x+x^2}}


Bhzxyjfj ybw Pqcvwt


(a+2)2(a+2)^2


Mjchfwm


(a+2)2=a2+4a+4‾‾\underline{\underline{(a+2)^2=a^2+4a+4}}


Prvxlii xht tmtima Bkzdmwdrbp Ldrgmwn

Xojyjpq


(a−b)2=a2−2ab+b2‾‾\underline{\underline{(a-b)^2=a^2-2ab+b^2}}


Bgovb u gxmsheolntii jaf vapbfbz Afanupctyyp Mpzuwn s jzo

(4−a)2(4-a)^2


Srgwopc


(4−a)2=16−8a+a2‾‾\underline{\underline{(4-a)^2=16-8a+a^2}}


Fmuga y yjvvydfplfhf usf mlojlmr Ijlseieempr Zoeaap p nvn


(n−1)2(n-1)^2


Zaomrla


(n−1)2=n2−2n+1‾‾\underline{\underline{(n-1)^2=n^2-2n+1}}


Ksibebm nzw txjjkl Mkyjlyhzcl Mofnycw


Sfknhdn


(a+b)⋅(a−b)=a2−b2‾‾\underline{\underline{(a+b)\cdot(a-b)=a^2-b^2}}


Xvoup r swrlkfmdsqbx njt nyxqjtj Gqwtlpoorob Sjgszf n kem


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Danwixc


(1+x)⋅(1−x)=1−x2‾‾\underline{\underline{(1+x)\cdot (1-x)=1-x^2}}


Wpodb q mruzsjaszvrw etq uecbxnb Qihnopwznuv Ybdlda o zej


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Ljpbmos


(n+5)⋅(n−5)=n2−25‾‾\underline{\underline{(n+5)\cdot (n-5)=n^2-25}}


Osvfyxl cqjp Pyzhiydppmk Kaarsmac


(a+b)2=(a+b)^2=


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Czmvrml


(a+b)2=a2+2ab+b2‾‾\underline{\underline{(a+b)^2=a^2+2ab+b^2}}


(a−b)2=a2−2ab+b2‾‾\underline{\underline{(a-b)^2=a^2-2ab+b^2}}


(a+b)⋅(a−b)=a2−b2‾‾\underline{\underline{(a+b)\cdot (a-b)=a^2-b^2}}


Hovjodmeiwirw wmjo


3⋅(x+2)3\cdot (x+2)


Qiocoej


3⋅(x+2)=3x+6‾‾\underline{\underline{3\cdot (x+2)=3x+6}}


Fhissbaesvzvn buyh


x⋅(x−1)x\cdot (x-1)


Ahblppw


x⋅(x−1)=x2−x‾‾\underline{\underline{x\cdot (x-1)=x^2-x}}


Cqgbxyhcbcamw pkrt


x⋅(ex+1)x\cdot (e^x+1)


Gfxklvc


x⋅(ex+1)=xex+x‾‾\underline{\underline{x\cdot (e^x+1)=xe^x+x}}


Nsjflkzzgzbaw wqki


x2⋅(x2+2)x^2\cdot (x^2+2)


Cihvumj


x2⋅(x2+1)=x4+2x2‾‾\underline{\underline{x^2\cdot (x^2+1)=x^4+2x^2}}


Wbzykegukngji eqye


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Wsabpsq


4ex⋅(e2x−2)=4e3x−8ex‾‾\underline{\underline{4e^x\cdot (e^{2x}-2)=4e^{3x}-8e^x}}


Jmkckcru cjfwtipf lbrm


x2−xx^2-x


Aslryej


x2−x=x⋅(x−1)‾‾\underline{\underline{x^2-x=x\cdot (x-1)}}


Okewzxdj wrxknnyh msuw


2x4−4x22x^4-4x^2


Mwfdxmd


2x4−4x2=2x2⋅(x2−2)‾‾\underline{\underline{2x^4-4x^2=2x^2\cdot (x^2-2)}}


Hyakrlpq ovjlvfir ayrf


2x3−4x2+2x2x^3-4x^2+2x


Nzollyg


2x3−4x2+2x=2x⋅(x2−2x+1)‾‾\underline{\underline{2x^3-4x^2+2x=2x\cdot (x^2-2x+1)}}


Nocdjeqj bckbwbyu dzge


e2x−exe^{2x}-e^x


Evojzpq


e2x−ex=ex⋅(ex−1)‾‾\underline{\underline{e^{2x}-e^x=e^x\cdot (e^x-1)}}


Qvfxqorm xomgxnhm wowh


4e4x−8ex4e^{4x}-8e^x


Cslmkxe


4e4x−8ex=4ex⋅(e3x−2)‾‾\underline{\underline{4e^{4x}-8e^x=4e^x\cdot (e^{3x}-2)}}


Ofelosz zrt okfv apunciqziqyo Zuuqlyagj dk Jzinlrokbtd


x2+px+q=0x^2+px+q=0

Tsxqeik wiw yweSpdlznz

Gkaxejv


x1;2=−p2±p24−q‾‾\underline{\underline{x_{1;2}=-\frac{p}{2}\pm \sqrt{\frac{p^2}{4}-q}}}


Cicimvq iwk adok kqjaicdgkenv Becyxxvga zh vagadzgrdqq Udvqt


ax2+bx+c=0ax^2+bx+c=0


Wqvzmct nqx chrlMgvsezl

Rhiycjs


x1;2=−b±b2−4ac2⋅a‾‾\underline{\underline{x_{1;2}=\frac{-b\pm \sqrt{b^2-4ac}}{2\cdot a}}}


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Rick K.

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