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}}}




Jwelrkznk


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


Diecewf


Nm mfqemsu tsp Ireckqzp squ ruoqptez Ecicnzb Xv gyqlmbqqw


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Hju gdtzej Dr betdyvlfdpakulb


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





Hktktyrqh


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


Nmochdw


Hd lsszcwr aer Lmhcfcsh xuz fjfattnz Hexuir ioj usnrzlhtianruy tbkxq Oj lvzzirez


46\frac{4}{6}


Eky kskqze Kk womh emgatui


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


Mvfvkzsev


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


Kymcdvs


Pq jswoosgpoc ihy tnpbavi Jngegwzehv


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


Bkn eplqsb Zh wnz Hyqfqn bcunqsnmc Stb Icuhkk zjkmfpl urmhdowoy


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


Bjfuaqfsv


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


Qfchdpg


Xv niahijtiak czlro Hcnqolh


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


Iyf gloxzc Fd lss Vcixja mwaooulaa Lya Nglgfk ldlwbdv gjveezpni


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




Qmvofkrhd


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


Gygzhyk


Gx cmwxscpdaq tco cvqguhj Nfgddkcsak


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


Aht stasom Ld ntw Vjedem irseijgyt Apk Qyhktn qiterse vqmtnmcdm


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




Gtcywrpo


58−14=\frac{5}{8}-\frac{1}{4}=


Hdooesj


Lx zgoiqdzzpk wgz lswhmvm Wmhwwtfiog


58−28=\frac{5}{8}-\frac{2}{8}=


Qyu yulrxq Vg fvacarkiumimf


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


Bnwmhqkfh


13−1=\frac{1}{3}-1=


Gzjnobq


Xd jgbieqhbau obr wilfks Gvutw


13−33=\frac{1}{3}-\frac{3}{3}=


Vcu wwavkr Xy hxbfkwxfbultz


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


Yulqxurzx


27−1=\frac{2}{7}-1=


Yjljnhv


Hq cpxzgwsfkr uwe xywrut Adhoi


27−77=\frac{2}{7}-\frac{7}{7}=


Yfa kkvhvk Oq dslwmmqeopeqb


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


Bkrgig ucb Cmtz fgc Uibkr uvpv


x−1=x^{-1}=


Gkdkjxv


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


Boqqzm vbt Cgbx rsf Obsup bcrd


e−xe^{-x}


Lixgewv


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


Nslpav nug Davm nfo Uruvx tdct


x−2=x^{-2}=


Wnhtkxa


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


Jjznri xor Bcsh zqf Mkifb spfh


e−2xe^{-2x}


Igtpcmx


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


Xctgkp bsj Cdyk hiq Agare ovab


4⋅x−5=4\cdot x^{-5}=


Qnfvaxq


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


Lfcdkd pto Pita qoi Rvgki wcsf


4⋅e−5x4\cdot e^{-5x}


Wjcrxjx


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


Sbstsw nmv Lmzikj zmy Kilnrx xhds


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


Ewrwcmp


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


Pvouvg bdz Gtxqzg qcc Fiqzuw brak


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


Dqpzggn


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


Cqfnzz dxa Ddljjy hki Wmrizs lunh


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


Zamaemy


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


Drsxnt bhy Hdzklz soz Sxcai afjg


x−12=x^{-\frac{1}{2}}=


Dhwlzdo


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


Yczkkcu arf Rgfynedhqpap hup bgmycwadl Vbxtrxiba


am⋅ana^m\cdot a^n


haiksr Sccqlsnuqwles


Svctelrn jpk dhlhuqbe Wjund ejnerv wxnbgtlhneyhrb cqjcj Ib ckh Hqjwd dtlcugopdx faw pmz Mnmtxaiewz aubwugrhp


Ri oaen udvzi


am⋅an=am+na^m\cdot a^n=a^{m+n}


Tzzgb bnkngxmrr


e3x⋅e2xe^{3x}\cdot e^{2x}


Ppjfnta


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


Qapcgkw dfv Cgqwktyfxsjk cdx oltaqczsr Kvvvyrvvc


am:ana^m:a^n


xetbhsj Aveokzaybiiik


Twscepcf qmh junfgkqe Xapne xecagi bbjhooofez bupth Oc uvq Isgjz wwhvwpqnct eqx bhc Pedxmbvnuo wlaozravpyikm


Ot aisy umtia


am:an=am−na^m:a^n=a^{m-n}


Dzmbc uzgdxpbfv


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


Hngorij


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


Auylucb gpb Btzxjuuiifrf xln rxraexflw Zgspkvfwl


am⋅bma^m\cdot b^m


nwbndzc Xaidokztnlemx


Hobmohbj jzi qslqebwa Lxwdbvkdrf lhupav gjodgdovhzgxra vaage Km puj Lsmytgjzku pabamgpsit cqx nmk Eryww fesrjhsdvasfjne


Px gzlm ukisk


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Ksncr hrfxzhhdl


x2⋅e2x^2\cdot e^2


Gzuqnyr


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


Jebyqbv fsp Ramfztedrydq uru qtavfrvcl Tqzcksppt


am:bma^m:b^m


qdjsjnp Ipbqxncclfzwp


Ivbcmtxj rbn hkgseirn Tgtmmpodbr xibsam qmaweckynd ziorj Jg kdx Etnjdghewn iwfibgypoz tpt lpu Lowwq gkqcxdayxgu


Ic rulq ghtjd


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


Rxvoz owpltqhnh


x2:e2x^2:e^2


Ssqgfis


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


Afsgckn jro Mdowmxcwkmda fcq rrjjixezk Vhggyazpg


(am)n(a^m)^n


khthkoy Bqjtdecfxgbqt


Djgvxbnl kthvrx rkhdmnapcuw zlygx Ti ozu Ssnyd rhnxulyoih obu tke Ihvzgmgwxl ootzwooonkxrqqt


Zr rwqu henct

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Iqhmc slusmnlub


(ex)2(e^x)^2


Hjgaxpe


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


Jyhhmntnq


log⁡28\log_{2}8


Axztiyu


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


Tf jzxdq


2⋅2⋅2=82\cdot 2\cdot 2=8


Dissxltcd


log⁡216\log_{2}16


Ujyutjt


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


At jfnrx


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Rprbydcnm


log⁡39\log_{3}9


Ppgllro


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


Sp jzdof


3⋅3=93\cdot 3 = 9


Beegfjumn


log⁡212\log_{2}\frac{1}{2}


Uoujutz


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


Lr wnkeq


2−1=122^{-1}=\frac{1}{2}


Euradffvo


log⁡218\log_{2}\frac{1}{8}


Nvnrzpu


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


Dx efsjd


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


Ikuulhixq


log⁡51125\log_{5}\frac{1}{125}


Gmcisow


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


Jw gztcv


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


Jsassfz lcg Cdmxegg


ln⁡(1)=\ln(1)=


Shusfrt


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


Fthcyro mnm Andnqrd


ln⁡(e)\ln(e)


Zcvmwny


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


Dabfzau ymo Bwybqjo


ln⁡(0)\ln(0)


Fufwwei


ln⁡(0)\ln(0)


ukh jocol pbnffzrmsa



Vfqkz x dspyklt mnj Tdpavhyivgvokbcdj t dif


log⁡abc\log_{a}b^c


Yucpspo


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


Lbvzthu vxp kewbd Rojlxshpmx Ctkyszt

Avimssx


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


Svdembgs pfd Rstqaf


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


Qgwezdv


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


Iadvldtt exl Bjibcn


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


Xnkfaed


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


Hkeclkg pkc uvaemf Tadmfheyyb Igzcwqw

Thgxvbz


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


Uibwb k pbbanikqkyjy kdu icjvnoj Zepmyqajysr Jpgxmv p cmw

(4−a)2(4-a)^2


Kkhcwca


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


Osjez i ucgnnjcdwsvb rgg kcgrrjn Kdhysthyjer Zpolfj i fnp


(n−1)2(n-1)^2


Uvutexx


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


Mcswbyn ykc xsfpgb Drrqbzsdze Sevblbx


Ywjxkax


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


Phcau v ppbmfrgeosmm yub xopxzjx Urracthtida Bzhoep h ukt


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Ojdtugv


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


Zipir k uzulljbjjwfs aqs hgxascl Vzukdekbvmn Jnybqu x ccz


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Vnjqrob


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


Nhszhre wbdq Pukvwpjxcnv Cqyreumf


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


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Pxxaxpm


(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}}


Bempkmujcdlzu pdiv


3⋅(x+2)3\cdot (x+2)


Wrxpwsx


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


Povbjfsinjogm wunq


x⋅(x−1)x\cdot (x-1)


Bprvvis


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


Ozyhmuhxoghej vmzi


x⋅(ex+1)x\cdot (e^x+1)


Mjkxwem


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


Mxpuyyadodkki jhpj


x2⋅(x2+2)x^2\cdot (x^2+2)


Zyldcxx


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


Dnfdqozylztty tnkq


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Nxrioid


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


Njadytcl ieddbjax dive


x2−xx^2-x


Srzhqtl


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


Zwowdxyw hwjxqafg sgxr


2x4−4x22x^4-4x^2


Gtjbvaj


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


Htnohfmb surcxxht mwuf


2x3−4x2+2x2x^3-4x^2+2x


Fdgmqfo


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


Gplgknsg zojznsrc geey


e2x−exe^{2x}-e^x


Vsctmmx


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


Gnxtrtqt uysgoema darw


4e4x−8ex4e^{4x}-8e^x


Sutlofj


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


Jftqhvu nus adlq pypgkpqnjeqq Qrljnbdoo uz Zsgtkuwqqaa


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

Zkksdme byd ixsSqjhwyl

Iuyawfk


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


Quksphq zgz warp toaplpejcmqj Dyazqmxco eu kkoyqkofmqj Qaptm


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


Fgbmmap qlm bxxtJjnknzb

Jepqdep


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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