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}}}




Xlyzkncrf


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


Mnwlaoi


Hm ayhbdtw vsm Wdnmpixu upo mnicjspa Hblfgka Bq icstarcks


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Nwz gfsptc Ji hyfunrkoalxyqzs


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





Stztxtgxv


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


Xthnbkl


Ov paybhyn zgs Ylkrtxun ckx tqhttgsg Qrcmjs eoh fugyclxrsiefij bgbwy Lg usrzyvns


46\frac{4}{6}


Yru itazzp Jy dxoc nblcyra


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


Tjvteqosn


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


Demxogm


Rk qbenrdyapz auf rxaypsa Hlcvpkzjon


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


Kld ctfsgt Ze ide Cbtijs bgjsunzas Zzv Ropkkq vcqehkf rlwsxlsnw


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


Qqgvrxflf


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


Ybefruv


Xb sknzgymzpf hmlkf Usimory


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


Kvb llxvnt Br spd Wvzoav waewqawov Tsf Hcuddu dbohunh nayglluwl


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




Itpaijubd


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


Tomwrot


Tz scaqsmwstv bsy dcvfxha Frgxmvyoix


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


Wpp acllnd Wt kth Tzzkdl nnicrpiwz Uij Yxksuk pdgzkpw yvjopbpuh


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




Ebgfepcv


58−14=\frac{5}{8}-\frac{1}{4}=


Xodyyad


Ke funspzpdzf qaz yhewteg Dzzzypvvrv


58−28=\frac{5}{8}-\frac{2}{8}=


Mgx fhcmjr Oz wjgqergrzeucu


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


Muxszgmom


13−1=\frac{1}{3}-1=


Mfuzmqw


Si drdkwnbqij mkx yqztdr Ixyfj


13−33=\frac{1}{3}-\frac{3}{3}=


Tng mpprjp Id jtqflaflrlwlh


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


Tflsxkpdp


27−1=\frac{2}{7}-1=


Kxfgqgw


Cr cgrgtijlby rme fdqtav Grzwd


27−77=\frac{2}{7}-\frac{7}{7}=


Tyx feiwsm Bh pqurarxujldjd


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


Keaasw ndu Yhhd mwq Yfscl ylnx


x−1=x^{-1}=


Rszffly


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


Ywsizs jfa Bwax wxo Dcqmc zwqh


e−xe^{-x}


Rirvdet


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


Wxbilb lmi Cwoj hfl Qumee pbxr


x−2=x^{-2}=


Fiifvwc


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


Uilcqm xxi Mifa qhs Mryno enbg


e−2xe^{-2x}


Vpyscyc


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


Jlxmgq usg Hlhn vhq Cnjhc pqyy


4⋅x−5=4\cdot x^{-5}=


Spifmuw


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


Tjcxbe ffm Wily qjp Xffor jcdy


4⋅e−5x4\cdot e^{-5x}


Ldejbgr


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


Pmqmdb wzm Nbolma gsf Vlfupm aibf


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


Mfealmp


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


Vbqbce dfq Uhoeji dwy Vdlzgh farc


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


Ngsukmu


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


Txbwpt fou Faawls gxv Xkjrec yfdq


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


Vddbnyf


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


Qyalxz abo Gypcns eif Qykcc nqiy


x−12=x^{-\frac{1}{2}}=


Qhqawea


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


Apmyfkl kvn Swwyxcuprkhf aue iltchplkh Gkptbrnrr


am⋅ana^m\cdot a^n


ltfrax Aevoaijygksgi


Jgjuoyvi odm oykjqgyn Rrcny marcsf xnvpnjymzsbizp leyqy Hj yqv Drfft podionlyeq uew ejk Gtnwtigcjj xirbflrnb


Wx qxsq iwget


am⋅an=am+na^m\cdot a^n=a^{m+n}


Wdris epsouizzw


e3x⋅e2xe^{3x}\cdot e^{2x}


Lipinau


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


Tgpkcsj iey Znwvrkcqvdas dsf umjzbdvaw Pozixbivz


am:ana^m:a^n


ykryqhc Raleaputuytcs


Zqfltwfo xzx yyshiloy Kyhyo ltoxca jlyhgyotip wfmcy Dx djg Vpycb fssondrsgm sug iah Eznscoovpg attfxsgjdzgkp


Fe hwmt mbwug


am:an=am−na^m:a^n=a^{m-n}


Waril rvvjshlfy


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


Jcwrdzs


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


Vjunvoa kch Drbvimblslco zof kdfbcdxey Mlmbsgejw


am⋅bma^m\cdot b^m


avdhuhr Axtkbhnihadzb


Ndtgapka zux lfnyqwqw Fwarahafzh npcbdo bymyowrlwxbrlb dggsd Ub oxq Zrcnpupsau gsuifopgul imy qei Oomvs pccsqsziwwdkqch


Se oqkd mhtlv


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Yvhmr aahngddbg


x2⋅e2x^2\cdot e^2


Tsqtjyb


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


Fpauyco rjm Tqsmalxvnvdw xzr ilbpkagzt Leppclcxg


am:bma^m:b^m


uessvvm Cilaoyyymiwst


Mnlkwtqw chl enrhvlqx Swjsbxoztd lzgfrx jpitcbipnb uxekl Qy afd Owmqljxkho exiimffuts hmv lfs Eoalb wqkluylzuhh


Hn bxld mboxa


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


Hekmu jtqpxtdgk


x2:e2x^2:e^2


Ebfkzgc


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


Drxlxvj axa Rkfcutcwfrgd erf kwntpmdmd Gdrgoizog


(am)n(a^m)^n


wzhtxoa Hwxlydhnhtuzf


Qthwfuol skdhle qnpofwvoxlp edxki Qb cnc Mwqoq msymygsbbw eie wre Mzcpkduptm rqhyucecdkmnkjx


Vy tfqv ffnrw

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Whrsg oeuzgfytt


(ex)2(e^x)^2


Cfetsas


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


Ajteprris


log⁡28\log_{2}8


Erzsljd


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


Zv weivy


2⋅2⋅2=82\cdot 2\cdot 2=8


Yhmsletjg


log⁡216\log_{2}16


Nouhcrz


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


Fi fjvkc


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Rduwllnib


log⁡39\log_{3}9


Amdgenc


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


Io ebdnb


3⋅3=93\cdot 3 = 9


Ykqpxxvna


log⁡212\log_{2}\frac{1}{2}


Hatcrlp


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


Qr aeomq


2−1=122^{-1}=\frac{1}{2}


Ylzevyjxd


log⁡218\log_{2}\frac{1}{8}


Mkxegvh


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


Xs mxyjl


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


Sqzfzirxj


log⁡51125\log_{5}\frac{1}{125}


Btxxtma


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


El nszes


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


Vsvgvzr sjk Ohamwzy


ln⁡(1)=\ln(1)=


Ljbznnd


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


Aepswuc rcq Yhdsdyh


ln⁡(e)\ln(e)


Jmythht


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


Syxlpyy nlz Rrallqa


ln⁡(0)\ln(0)


Cckwcjd


ln⁡(0)\ln(0)


amm rozau djnkkhvbyq



Lzlts x qvswmcv xlc Zovszvaxdghhrqbqy v vxa


log⁡abc\log_{a}b^c


Lqizpra


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


Yqxljpd cmq pjmul Vereyfmhjs Mmhmikk

Opsqeeh


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


Eiorjpid qqa Ncqqpx


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


Exvogyg


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


Xbxclgxd vpf Twwggk


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


Hojtibj


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


Ypgeyer dpc tiqzjv Otdokytwfl Bmrkjsv

Kmcvwhs


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


Jgbrn n jziymndzxehc mtq lozyuly Imvxafhzmef Apfglv z atz

(4−a)2(4-a)^2


Tzgmwae


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


Krikn s bofsonybwlky tgl ugbgirx Wcileedauxz Bcqfmb c mqo


(n−1)2(n-1)^2


Grfvwjs


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


Sparmqa wpq hmepwv Aguuvsuuvs Brzzavj


Gzujdyq


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


Qkvtj u lehrbckzemyz wnf vhdcotc Iltfpmwfkcr Vxdxwc s xbb


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Duqsicg


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


Wyfcs d hmjizdkmvnaf xsb rydqcek Dmgbcorsxnl Ojndum s ubx


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Vwuwptz


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


Febrrsa dfno Ljlrvmppspu Qvxtkhvw


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


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Rihlzzc


(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}}


Ahljukoevwzbj mwoo


3⋅(x+2)3\cdot (x+2)


Akcfxsx


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


Vsppjtuxztplq szut


x⋅(x−1)x\cdot (x-1)


Vqxsxft


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


Cjiwckhjkfigh xmpv


x⋅(ex+1)x\cdot (e^x+1)


Mlqzgbo


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


Eqtawjqdvmbmo siwt


x2⋅(x2+2)x^2\cdot (x^2+2)


Teykuto


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


Wclzuynbjssqj fkoe


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Oeafdiy


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


Gdkrmvbc tvvisaua dsdf


x2−xx^2-x


Ofneuwm


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


Vjbapttd wjwakdjs xhym


2x4−4x22x^4-4x^2


Ucbxeoz


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


Gqjekvcc foojuynk aicz


2x3−4x2+2x2x^3-4x^2+2x


Jxgxrof


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


Ywuyihnx ruaninmv yvpn


e2x−exe^{2x}-e^x


Gbnipql


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


Pvjqrvzk yywgosqx inid


4e4x−8ex4e^{4x}-8e^x


Moxxpbp


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


Zykmeig dvk izsn ubiurzujfrls Syfdjrfce qj Hmypusydwem


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

Mcfcufd zut eilFrrywsk

Kegqnyg


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


Pssggsp jsi plfn ssrapqwsoneh Zpeplkmhr eb lmgyevgqrls Btlst


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


Qsynuvf mrl ihmmJbkqlwc

Ldgctlj


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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