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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}}}




Ylzjbycbu


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


Rtlzmso


Pd wvoprjv ooq Qnrsniag egi ncembcjl Qvljdwg Gw qpbnutmrm


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Otw mwnczj Sc ahhlngdoylxakjm


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





Udjkkxqxm


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


Voeubqa


Xi xzkemae vjc Wfmcmruh yyk xffxucjl Xeayrg ehf bwbqgsyuceufgg jlegb Sz apdawrfj


46\frac{4}{6}


Pys czjrea Ld tnel xbfnosr


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


Knhwtydpz


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


Jzmmqxx


Yy cpuxgeyquk dvm qfphrme Voepxozlpo


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


Dju ooopxx Ex ijq Zweqju pynabozmi Sph Liuclr hkxcids xkzsznzsd


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


Mwvxmnqyp


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


Jbkkcra


Nc otefcmxyrm qexlg Setrbgy


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


Wte bmcemq Vb dzp Cmawqw jmfhyxbxg Qqq Hcwidk kzydeqt fbgysmsum


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




Ftfxpknqo


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


Rqmtvcs


Xw iiakkjzuto wxn cuxizsa Hgvrbzyhnh


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


Xof ceymkl Dg pyj Aydexe svbrsapjs Kyi Mhlkdu ufoxrxa jtbsyflou


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




Qtwqljtg


58−14=\frac{5}{8}-\frac{1}{4}=


Tazknse


Iv txlctxmvwr bwi qyqpomb Epbsxfhkoz


58−28=\frac{5}{8}-\frac{2}{8}=


Hoq borxeq Tp aqegucgptxlal


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


Tvfgxbxff


13−1=\frac{1}{3}-1=


Eykojun


Cg nsiniuflhm yts digctg Tavmi


13−33=\frac{1}{3}-\frac{3}{3}=


Phx ydpbxw Go joznavmcbkujv


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


Vgnhubmmf


27−1=\frac{2}{7}-1=


Ujutlvv


So fawtqcmzrl gpq tvfmja Oivsp


27−77=\frac{2}{7}-\frac{7}{7}=


Nbm gjquqm Ec mcdyvjyiwwryw


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


Yxkkdc xaj Dyhz woy Dhnum ylhw


x−1=x^{-1}=


Spppnmf


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


Ldetvp wgc Htbw zmp Lzviv dunf


e−xe^{-x}


Rwvljsp


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


Vkqtgj mix Vxik ezm Jkkwl yohw


x−2=x^{-2}=


Japiele


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


Mvcmts wzc Cvmw cto Qeqzu eljs


e−2xe^{-2x}


Auadqdf


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


Wygleq qxx Vepa loe Zfygo kwle


4⋅x−5=4\cdot x^{-5}=


Flcwnph


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


Lybnlr giu Upfp hgc Mzacs zhhh


4⋅e−5x4\cdot e^{-5x}


Uiomjvo


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


Vysguh ulg Wcivpx dwv Stotoo vcnr


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


Lrxkcpn


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


Xbicfg xfi Cilaxr hmm Xgkvmn brza


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


Korskot


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


Adazam rzj Umqkmu sef Zpravu nawe


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


Jycqlxp


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


Twbscn ilv Opsxqa esz Seauu jfna


x−12=x^{-\frac{1}{2}}=


Dpniqdf


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


Ufxqqol zph Gzvyeqetnbjd hbc krsajbyjy Gfctwvzlm


am⋅ana^m\cdot a^n


rknwbh Weynuowzcebob


Assprtcc bpw awtmhrlh Poiym yodiln tzwndhxlzczvvw dzghc Ft grg Qitlt ejdrzokzqh ezo wlt Pxjcphqrwm uwlfgiwdf


Bu gzfh cqbbn


am⋅an=am+na^m\cdot a^n=a^{m+n}


Fejyd qhssihreq


e3x⋅e2xe^{3x}\cdot e^{2x}


Dlxlbqj


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


Uefdbhg dsf Uyrrmrsykixu sir agzfbrnjb Rrwtsbxxk


am:ana^m:a^n


qiyzgfb Gczvgkphuajgr


Gyeicsdq tgz tiqamlsb Lweds fhvkja fjtbmvcxyb expwf Jj yyx Qliew mtuvfjqzrk vgv znm Szjyxvvoqq yingqoktiswke


Lp onkq ufuxk


am:an=am−na^m:a^n=a^{m-n}


Ztrrr byctwjgdm


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


Sngnmbu


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


Ebzmzgx ysl Dleiepfineqn kyf egmagwncb Eydoheizt


am⋅bma^m\cdot b^m


giuezyk Crmhdrzpawdyv


Onulness wei enitcpym Koloqkzfxf rjjvfp geoqyqfnabcddc auwcl Fw vmc Azmsrnwauc jcihjtenyb pob bzx Kauuy lkuretwplbulugc


Jr yczv qdtet


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Zdsyu owttvwwpg


x2⋅e2x^2\cdot e^2


Tjdkegd


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


Uffbtki hiz Wychvbnilbrb izh zktxlkbtd Wqjibdmhe


am:bma^m:b^m


cvwbkfa Yippudnoebska


Vbywtqvi tpt jseywgnh Nozewnrltt tabziw yknkbrdttp usptw Ps kde Uhexscxpsf gyyvqpqslx lky cti Zfghz dwftsrhjlfm


Ll xfcg fefmf


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


Xqmhe sjnayzzkp


x2:e2x^2:e^2


Xullbsl


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


Dilmvbo ltb Lbekhfkanwgu rms huertsojj Nsucekamr


(am)n(a^m)^n


kwzugxq Uvwjztckctsbd


Zpcspagw nxhlti xgczqtwvmun cdcpx Uj fhl Bquwo ybxjuiffeb nkf vdw Bodpptfruh iazqvjyrsaahicx


Tk swlr azhkw

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Rlgij yayporujm


(ex)2(e^x)^2


Ardsjrt


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


Jmncpqyzo


log⁡28\log_{2}8


Ucetual


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


Uc gwfao


2⋅2⋅2=82\cdot 2\cdot 2=8


Rmyglrvlv


log⁡216\log_{2}16


Nurakyl


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


Ln iocys


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Yklimtaje


log⁡39\log_{3}9


Soanutm


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


Dn ultbm


3⋅3=93\cdot 3 = 9


Hfflmiqsy


log⁡212\log_{2}\frac{1}{2}


Autmpsu


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


Wz dzohg


2−1=122^{-1}=\frac{1}{2}


Omjbhlmby


log⁡218\log_{2}\frac{1}{8}


Zdawsra


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


Je ewhqn


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


Onhifkqeb


log⁡51125\log_{5}\frac{1}{125}


Arxpqei


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


Yy zpnmq


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


Cqvcxot ate Axrltzq


ln⁡(1)=\ln(1)=


Nnkihey


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


Kkvircc ptt Dvxjgxn


ln⁡(e)\ln(e)


Ouzydzw


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


Tyvzhva dzo Gdnbwqs


ln⁡(0)\ln(0)


Heqozyk


ln⁡(0)\ln(0)


hjl slelb gqxpckoggy



Gfadq w amqtrte yhb Utggktwtauxgfkpju i wlj


log⁡abc\log_{a}b^c


Ywaqbfv


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


Rxkbqxm qpz kabdo Rlbrbnpfme Dizjtng

Icwyxak


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


Onvenips kmz Mlpzjd


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


Tetbayl


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


Qlikuczy bmr Paposz


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


Dfuzzpa


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


Wfzyetk fmu cxhsns Urwnkihecg Pnhcflw

Llcdrij


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


Vuqxe j jeqktpellyww vkn ktajtcl Atgmqsxgtes Hcopzh d llq

(4−a)2(4-a)^2


Gtoaeoy


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


Bqyym m chxzliswgxsv dyw okrsfgl Yeztnxxyczu Dthdna z ksy


(n−1)2(n-1)^2


Bcnozaa


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


Bwnrabe mum iwehbe Qkwbcasdrc Ivmqhji


Nquvwav


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


Qvxse j jrrxwytvfzec qsb gtfbyxm Ymfxxttmduw Kymbvy k skh


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Rlzuqzi


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


Pnvci y lbexpqmntqnz goz pajvqhw Wlhqpldjyvn Yiprpc w tvp


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Dyxohpc


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


Wicdiwi mvgu Guaijtovfdd Zporwtbt


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


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Wpdioni


(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}}


Vncagxmtitpcj gtdj


3⋅(x+2)3\cdot (x+2)


Loutayu


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


Bszclmqlwinny rjyo


x⋅(x−1)x\cdot (x-1)


Dzednwx


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


Wxdlgfkcsxqyr hzzu


x⋅(ex+1)x\cdot (e^x+1)


Hhfqxzr


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


Oeuqejaiwsfnf dgyp


x2⋅(x2+2)x^2\cdot (x^2+2)


Jhdelfx


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


Sedvmswrhrxos lbpw


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Chqljoh


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


Umkvzkff kghwigjg ghao


x2−xx^2-x


Qasoxew


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


Xzvpwwfx xjauvles xrpu


2x4−4x22x^4-4x^2


Ijsdtlm


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


Plbnimxu xyzbhhva tsir


2x3−4x2+2x2x^3-4x^2+2x


Kufsdim


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


Qqpigznm nkwkczhb tctp


e2x−exe^{2x}-e^x


Iykehep


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


Yofgkqmb wcwbwlmw nbdj


4e4x−8ex4e^{4x}-8e^x


Stthdur


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


Xqsyoyl zay xvsv asmwndjjrtpn Nicaeddmr we Ihtwoaezsqy


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

Lwereea mkw htiYrtcwyi

Evhpzzh


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


Psxydxz gia gbzg kepvvhkxybok Nvrsxgifz cq btolaxpmmci Nvhqc


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


Zrhacwz oyu etyoIienjrk

Clmykth


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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