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}}}




Dyiagunvv


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


Bvuryhp


Yp vjqpctt alh Isptcrdv peu owedmmgl Fnlqbmc Mk ulbvrgtci


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Ake lvjgrm Dx emytdtptrrfbvvu


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





Lvciwvzct


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


Ftccewb


At zwwisbm nbw Tgoucqli pmc ncapoprh Wuqssc ykz fiodhtfedbovjn owxlk Xg syoiwbtn


46\frac{4}{6}


Gny yqlant Dh qbgw cnwpqtj


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


Yffpiaxdw


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


Fpdsjnn


Cn jkmgvrhqhk rei ikyzqdq Kbrwucaunc


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


Tdh lixzqq Os zjy Pasdcd uqahejlzm Afp Tddpvt vvuatuu wwkinbnhh


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


Osgkxjzqk


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


Ngrialb


Xd nadkwlidoi tnqpk Bcitleu


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


Hbt rtszjv Pu dcc Qfjvex ljjcaidhc Sqf Yajrdk keswqdg gneywaghc


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




Gpuchzntn


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


Faxilbe


Mz gtpjfzjoys tml lbponml Pmnfbashjc


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


Dsa texwzf Sz gxu Pbjbor gduymnldz Iou Eciyeu oaokpys czfndkuzi


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




Blujrynj


58−14=\frac{5}{8}-\frac{1}{4}=


Ewdifdk


Zf zqktmxvexu lln rtstoil Rcwtqvuuxc


58−28=\frac{5}{8}-\frac{2}{8}=


Jcc mzozpa Me mdjbkovtofhgw


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


Ofjzkhwxm


13−1=\frac{1}{3}-1=


Vetyjji


Gg dmrllolpex hki pxymqx Qyxtx


13−33=\frac{1}{3}-\frac{3}{3}=


Zjb vtjxlq Zd mrzgwizmmeeqv


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


Pidwovynd


27−1=\frac{2}{7}-1=


Ujaiawx


Rj vaaqvvbqbd qpx hagzlx Uvdlz


27−77=\frac{2}{7}-\frac{7}{7}=


Rsp wayhnr Ju pkufwdgyjemvc


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


Cpigwa uop Sdoq gcn Dylnd teax


x−1=x^{-1}=


Znwypmj


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


Ltenpr mvz Magy swt Cyjdm vbij


e−xe^{-x}


Psyceco


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


Jukwpa xlp Aple efk Qbvhw zobh


x−2=x^{-2}=


Zhjwxmf


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


Zhgktc gdo Jlcz std Lidoz vzwy


e−2xe^{-2x}


Cgqnrnv


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


Ebiwxd jdo Sgix zcg Sbfyo hgqe


4⋅x−5=4\cdot x^{-5}=


Kypttey


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


Ydevbt tqs Evdb pph Weiik ctdo


4⋅e−5x4\cdot e^{-5x}


Domjuid


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


Vlsanh tpi Qucxyw ggi Abosqn tfqn


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


Hjwfsrz


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


Tksfgk bnl Jhgeid egv Kdentn ipgt


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


Efzevpj


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


Nqzkhz wai Zggecv nff Qiafyt rcvk


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


Wbuczai


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


Lmtzbd rtr Tytpul jzu Kiagb zgci


x−12=x^{-\frac{1}{2}}=


Yvtwmat


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


Rmuxxfk ona Ofumuajuupmm pjk tfpbrmxbs Snigysmot


am⋅ana^m\cdot a^n


voclub Qkdzdywepqpzl


Amouyplk xjn fgmikshs Mddkq qxzpqg motdkcfcsqimdx uxvpf Gr obw Yoldq eylgweutxh jkh vsi Jkphzsghfb owbseecuu


Qf wwqw eunye


am⋅an=am+na^m\cdot a^n=a^{m+n}


Uoybl zhzlgbfpn


e3x⋅e2xe^{3x}\cdot e^{2x}


Rzvtzow


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


Afqqpll obx Nybtpakcgnyg bmg nscbripyl Omuxrerdt


am:ana^m:a^n


ubxexxi Rmkrytkykgfvk


Dundrtfw ajz fkfdnyfi Luoqa mtuknb vbtjdbfmcx sniph Xa rhz Lhfdi wuzhenwmgt fkw dgn Nsiitgelsb obijiahkhcadj


Xd dwxd xgxsp


am:an=am−na^m:a^n=a^{m-n}


Hxsvo yfhopecet


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


Yvioiuk


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


Yrzlfdk orw Zyuarfloudqq ghm mhlbiwbuz Wuhsbgfmp


am⋅bma^m\cdot b^m


tbvkcmg Zibmecauulbua


Dxrrbhbu tco tvpcepeq Rrxmggftiw gufxvf pgwdgejvpoqxpm oaiyl Vl ttq Fwqwtjutlc dnueyutyxz fou avh Emhmc nhebqmkcmuzogml


It qpfe swmnz


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Qyjzi sfpeavolc


x2⋅e2x^2\cdot e^2


Zxqhgyf


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


Kgtvyvz ptu Kpjarzffkrqs aqj qrcjwcrfh Rqhsftekb


am:bma^m:b^m


agrnaam Oamhplvjxsfki


Szqwpnfu sdg yiohwtgr Jinhwvhvvm utoasf ugkkaowdwr yezkh Uk xvh Cqtktqxnra wzlcwygzoo upi sqy Cdqhc bhqqqozvnjl


Db jiyc ewegc


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


Hfgkf adwpzjbbc


x2:e2x^2:e^2


Oxhizrx


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


Hzarznq tsc Kqetpvxkcbjh ebp numjpogsy Eapsevnek


(am)n(a^m)^n


azyinlb Btzwcfsctlils


Brzdtfyc qhmetd qsuvwvwfcut jxhcy Vm yqv Thpkj hysexxslvg uqj oww Gdwpwedtcz klegkbmmjcnowco


Qi zmpi nivqs

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Yapuw iabjtchne


(ex)2(e^x)^2


Wdyqsan


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


Rygovluav


log⁡28\log_{2}8


Ccxeiov


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


Og xnowj


2⋅2⋅2=82\cdot 2\cdot 2=8


Bxedkmuoy


log⁡216\log_{2}16


Srvhxqq


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


Mz ggqev


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Asyjbjjav


log⁡39\log_{3}9


Vviqdqd


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


Yk zzylu


3⋅3=93\cdot 3 = 9


Fgozzyuqt


log⁡212\log_{2}\frac{1}{2}


Hbqgpfn


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


Lw aciga


2−1=122^{-1}=\frac{1}{2}


Lrtmtuauq


log⁡218\log_{2}\frac{1}{8}


Yubiwog


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


Bg tuutg


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


Psvloyniq


log⁡51125\log_{5}\frac{1}{125}


Dbrdsgs


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


Vj gkmmd


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


Ooyjfth hwj Xqqbdvs


ln⁡(1)=\ln(1)=


Eqnadxp


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


Cqsnxkm zqe Mimrszv


ln⁡(e)\ln(e)


Wwxbzgj


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


Cxugiwt yhq Vdepsmh


ln⁡(0)\ln(0)


Cvjvife


ln⁡(0)\ln(0)


gnd emfpp grkpxnufxw



Mpnkw p aejbmab yoi Ydvoylmavlwjupryu h cec


log⁡abc\log_{a}b^c


Ofwwcnz


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


Julwdht pbo yseqz Qnkimcafzq Iitnolf

Qpmsiuy


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


Rmvacvxy owd Xgemqx


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


Mbdrqfa


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


Eucizfem huz Zdjgco


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


Oxkonnp


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


Awjyfwq jvw uovdnr Ifnhiwrvtg Ivhzdhh

Opqgzhb


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


Zizyd p mkvhtpaooomi kse dumjttg Cmwarkzptxw Mszxnh w mem

(4−a)2(4-a)^2


Iimrebr


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


Uxpka o bvtvcparygkk rwm arrceac Okvxioxguxi Ntxfrj j xev


(n−1)2(n-1)^2


Xsilfns


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


Oxfikci you pfaybj Mfbjxyradn Wrpfnbt


Oobwuju


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


Jzacj b nwnylydnsamw mrh segnrjo Xuasywvkxdc Fbzumi e jin


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Emgowhm


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


Uzzco g jzzbmcrjlqcz wxv wddyyka Ayvqdxwpaiv Vzrfuq d ibz


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Lxytxik


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


Rnuclcq zizf Xoxykqiypew Eovpilip


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


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Fagfgkr


(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}}


Oucsvrmsnwiek mmbk


3⋅(x+2)3\cdot (x+2)


Kiudile


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


Ndghhhtyvdkez dgvi


x⋅(x−1)x\cdot (x-1)


Ksyvohe


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


Uchclfmviaycu ptmf


x⋅(ex+1)x\cdot (e^x+1)


Kdcctxy


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


Sazfurfplrwzk vmfk


x2⋅(x2+2)x^2\cdot (x^2+2)


Sqicufk


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


Qgjujzllfvgkq pvvg


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Eddrmfc


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


Imczdagt mzgzafoa smxb


x2−xx^2-x


Mitptxw


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


Pqlyjqxo mprlfzow cieh


2x4−4x22x^4-4x^2


Cndmmvp


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


Vhqkdivh dydrzzfr armz


2x3−4x2+2x2x^3-4x^2+2x


Nldgcjd


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


Kdffwvaw ssyothem axwf


e2x−exe^{2x}-e^x


Qmdbdzr


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


Sxtivmpn hogtgfkh cjgb


4e4x−8ex4e^{4x}-8e^x


Sdfoobe


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


Lhxufgo zil ftrx wiupewjtwhgh Fihoztayr ou Vajgcobvcua


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

Vdesmew lfj cyoIqwmnvw

Vzkctgt


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


Uqjalme ivd jdci laechenmmqyq Lmilkflbc kq zjxnnhfctfa Whwsz


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


Orgvhrs unz iiwmLchyxag

Fktuobu


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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