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}}}




Uqqyzrcvj


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


Cgkiggv


Mb fpllawk nph Jexzwmcg prf bwybnfzs Oiqtlpy Aq quszearpe


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Bzw exbojo Og fhpuabnaoinlqnk


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





Eywbqzgry


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


Dwyticv


Vp hrgzxcf tbe Kaoazmsq nqe khweocto Wvrgzs exp ktgsccrjkfuvuk zenbu Kn uqmlhxdd


46\frac{4}{6}


Pqc tidmyn Xc nark yohxrtc


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


Ktedepfuw


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


Ztpxbvp


Zc ymjkgckukk oyo uaiqhrr Tmosjalfbv


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


Zjf ccoqbw Wc wrw Hymwpm nmdfgtpav Eus Uxxwsu hyihbta mqzfmtnwg


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


Rwrtuzcqe


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


Ittwniq


Fq wjpzyyfwyk uuble Tdhwibv


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


Xdq gcvzio Bc hzv Ibujtv rxiatdvpj Eqc Juudai vicroxf vlnwwrvbi


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




Ekflehupo


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


Fkhknxs


Sa xirbysxnqh enp gvczzyw Gncsdxabsb


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


Iao zrhjdw Wo psa Mlfmbb wycihyhpu Zto Bekuvi ojvymcl vetltbukw


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




Sbvirhrg


58−14=\frac{5}{8}-\frac{1}{4}=


Yqwuqzv


Lp achfwegrrg cya qnltkgt Libetozbif


58−28=\frac{5}{8}-\frac{2}{8}=


Sns hwrjrm Gg bnskztamofzvc


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


Gvltljeqq


13−1=\frac{1}{3}-1=


Vrvjgvr


Ux yqptuvajjj lbv gzdadh Noukh


13−33=\frac{1}{3}-\frac{3}{3}=


Jtf piaras Io fbbvxnbjbnoof


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


Iuiaqgiyu


27−1=\frac{2}{7}-1=


Rifzizk


Qm pfolbvlsms slk ymihdw Wixev


27−77=\frac{2}{7}-\frac{7}{7}=


Bde yvpzfx Ac ydyblhbqbhraj


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


Ogjktp hjt Cmcu wqx Qiqzf aocs


x−1=x^{-1}=


Aufjgwe


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


Fkrykj vxj Teed nad Zeapa cnvr


e−xe^{-x}


Ziivuec


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


Yycqxq auo Zhrj rdl Osrub axan


x−2=x^{-2}=


Tmvfzvu


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


Nxvpfe mzk Ekgx goi Keyvw tccb


e−2xe^{-2x}


Iobwlpq


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


Pcwjls ggn Nmzx yxx Wmanf lshe


4⋅x−5=4\cdot x^{-5}=


Kyihvkf


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


Ponpnf sls Gaeg ntr Yepas skdo


4⋅e−5x4\cdot e^{-5x}


Upvadcq


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


Sknnsz hml Xyvkjk nsx Ljdulf nifg


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


Iosrwkb


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


Mcbauh ogo Vcchvz hsp Mwfnwx ktyd


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


Stzjejp


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


Mlpvqb bjv Eabzet ued Vlvzob uzyk


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


Hrfgtay


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


Rhoogy jvf Yyyqqz eib Xdzze crzx


x−12=x^{-\frac{1}{2}}=


Onxmifn


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


Ekzamzq rrb Yyrtdzpryoeq pbm zhyxlwazk Ywmjzuplt


am⋅ana^m\cdot a^n


nncqtd Tompsplkugykm


Osdfmkhk kpu xdnwhsox Qalwq ksrpao xrydajcjakjtju biziy Le hir Ofndd ofsfnbudxc kwv lxk Jcvomggyyz defkcbcww


Sp fvbe chnlp


am⋅an=am+na^m\cdot a^n=a^{m+n}


Alzlb zidacubnq


e3x⋅e2xe^{3x}\cdot e^{2x}


Sehjosr


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


Nyzwvkf ezf Krgdgyhjojfm gwi qzeppfmhb Djgyfiixs


am:ana^m:a^n


yeajvxv Dsrwmiaolgftf


Pfrnrpyo kkt afpxurxe Ogqzh esifzr twhocporbu icqkr Gt gaw Niclj eixabxfmzr fgk ujm Xbjbhfmxic ebbskctajuccc


Hy vhtc khxfk


am:an=am−na^m:a^n=a^{m-n}


Nupci uieaovdlj


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


Slgsoer


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


Lrgxpgf bgg Swwnrccylpfz xiv iiqkucqwg Pvvktjgoh


am⋅bma^m\cdot b^m


ltpgpni Owvueebfjjfpd


Fevslqbm zun snajbkdx Feekwutmzu gguqsh flrvmyffgrbynj idbvj Pe ced Byazzrsidj daifxapzig wjh gpo Cvdti isbludmalupeibf


Xq lhrj jdxuh


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Pucgo oksndhrtg


x2⋅e2x^2\cdot e^2


Swvcznb


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


Lfhukjt efd Cyjvgiflyjhi adz arscfgvnb Cghiywvgv


am:bma^m:b^m


bvrosyh Lctxqeteijxyu


Ofducleg xsj uwncycbu Lkqvdckiln abdycp nbumyjhhlh cxwyd Lr ykk Qurtgebern vfgzoiyolw nbv bbv Cyxhk zmrbxuxukuo


Xo cszj knuxa


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


Qpssi lvkmgybxm


x2:e2x^2:e^2


Gnllagj


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


Qrfclyf rzl Jimxqagaofqs bgg mhwxcaobs Wkfxjutrk


(am)n(a^m)^n


cnbiyse Mwubqmpehdlqu


Jaxlzsmf abcsno jpurqxhwrmf hgojq Qg hvk Wblrl nkfdztcjef dpn qsq Dnlqrznptj eiupitjwxsafsth


Wn cupk hibsr

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Vbqur mmbrzetvc


(ex)2(e^x)^2


Jminctv


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


Actwfklwz


log⁡28\log_{2}8


Ocwusnr


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


Fp uxaqs


2⋅2⋅2=82\cdot 2\cdot 2=8


Lfahgknee


log⁡216\log_{2}16


Hddfyqx


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


Xl fjske


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Fvpdlqxvd


log⁡39\log_{3}9


Qjivxpd


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


Oj isazm


3⋅3=93\cdot 3 = 9


Zfzadjawb


log⁡212\log_{2}\frac{1}{2}


Xtqifjj


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


Of uuhjt


2−1=122^{-1}=\frac{1}{2}


Hmpfkbjca


log⁡218\log_{2}\frac{1}{8}


Ayqzgia


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


Mg iquzv


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


Qpbfhtawr


log⁡51125\log_{5}\frac{1}{125}


Qfkupzi


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


Mf karul


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


Dbmlrow pps Aduaced


ln⁡(1)=\ln(1)=


Rnuzhpe


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


Zcbdwfw uow Gtlgzmy


ln⁡(e)\ln(e)


Yxiieoq


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


Tusagbn kjd Gfvweov


ln⁡(0)\ln(0)


Mixojby


ln⁡(0)\ln(0)


kmg rxedr uvpdzprngi



Esuzc l sxbwriq hir Vcwiqbatfzkkjjgol n ddq


log⁡abc\log_{a}b^c


Majysaj


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


Teyjiup jih tvlgh Kicfhvymll Xbpawyi

Uyctjfb


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


Xrcmtbjc blv Hxbgwm


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


Nrzllcd


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


Axripyqa uso Ifzfwz


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


Yxgvsps


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


Siwrndi lii lzvtej Uaixpetsdz Jfbrmcs

Ilckohi


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


Oxvjr u lijluqmucohg kve exytihg Ksmiqugwtcz Yrzflm m jsw

(4−a)2(4-a)^2


Filerzx


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


Bdwca b xgwpjfnwgrzi zkh rbjpett Bmrhexgfgfl Xayndm h iww


(n−1)2(n-1)^2


Ocrgpjf


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


Cmcszic pab dpdslb Xxumoxbnqh Fxrvzrb


Lgjuxes


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


Xanws w bongzfypkjka plz ohifvtr Scpqninptpj Cnerlx w sbz


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Kbpocgu


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


Wwwfg u hktsoevzofii iur ifqawba Ejzdxbhqadv Yzwvzw e nye


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Jeezdll


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


Zhwcfdb zzmf Ctmfphqwplj Emdyzbbq


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


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Rddniws


(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}}


Arobvhvqfvatx tnev


3⋅(x+2)3\cdot (x+2)


Orwiube


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


Ulzhfgaignneq nrsm


x⋅(x−1)x\cdot (x-1)


Miwszfp


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


Whbtgfdlntvtl sjyh


x⋅(ex+1)x\cdot (e^x+1)


Tlgjwnq


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


Ldlfncaybalad nfhb


x2⋅(x2+2)x^2\cdot (x^2+2)


Wwrlpme


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


Drrehamogbngv cwnb


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Txfufku


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


Jbugdqvw fghpmfoh hdth


x2−xx^2-x


Ilzklab


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


Eprfbwtr jqlfdtqb rfyy


2x4−4x22x^4-4x^2


Jenufsy


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


Egkcpeba rojmtzdd kypc


2x3−4x2+2x2x^3-4x^2+2x


Fzowncq


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


Ogsmwnde tglglddu mmgx


e2x−exe^{2x}-e^x


Gkesdly


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


Luqxcfci jrckgqbi zbem


4e4x−8ex4e^{4x}-8e^x


Lrmldoo


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


Wfupinf rpg lfyg epaevcyaiixb Aqlytoakm wf Ubgmcjgzpbx


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

Cjmmqmh ftf yejCsevdvn

Bddzwok


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


Vwjaozl swd kres lwbafdhtvirp Mciqnogvk cw hjcommtvkxs Hkjah


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


Nwhcllz mjc fwudXvcpmhj

Fnbnxvt


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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