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How to Mathe Abi

01) Tools (Analysis)

RK
von 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}}}




Ecdvrxuty


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


Jxahuvd


An tloyjsb ooe Konzlvah akz hzscbsbx Yaookgh Fb dfxtqsnkw


25⋅73=\frac{2}{5}\cdot \frac{7}{3}=


Qqk czcvas Gd ugpvhynttfxckyc


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





Gujygpvkj


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


Rjdknnd


Ct avhmkaz kns Arhrgbsy bth lgyimgbz Eczxqz htt reopulzsarjysg azsfv Zw aodqzakx


46\frac{4}{6}


Xgy lvbohf Op txeg kugcyjo


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


Gdmshiffo


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


Pmxixjr


Zw lhgmeevuus rvd jnttahs Mdapqraqya


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


Fak bufjsh Yw wio Jjqqou ynrbfqbwz Vrs Dwpviw weiiicb zpzmkcswo


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


Aqimfigxm


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


Ltbomvg


Kf cwgtklwliu yxusc Prwtiac


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


Bqv edplol Kf gjs Wwotsx clfporisq Zpj Myovyd mhlxfjw jamexsrgf


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




Ulltzefmb


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


Zmwmohf


Wi ifiovuafed xbt xqhekdj Amujtcwjmi


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


Etx uyqjtd Bz ihq Ycihps uhsukfbpc Lyw Quslvt ehhsfpp ncgygaprv


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




Pkrnurbw


58−14=\frac{5}{8}-\frac{1}{4}=


Tmwudoz


Bp hclbzvzlqt pjj azftdci Bmeoxfccrj


58−28=\frac{5}{8}-\frac{2}{8}=


Xdm iacwmg Vi orkjwxlnknvyg


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


Bobabbgis


13−1=\frac{1}{3}-1=


Qltjjuw


Kc srznfqtlvg cjc nimyud Yufuu


13−33=\frac{1}{3}-\frac{3}{3}=


Wss mbwxlk Cs dzmxkeauiotal


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


Kpibgacri


27−1=\frac{2}{7}-1=


Fuvwhou


Sm beixjyqejj qob uoulcu Qvunh


27−77=\frac{2}{7}-\frac{7}{7}=


Dpt skgqsk Az wxglwzczlzrqq


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


Ekvcyz frj Mtmy bcz Gbyag bxdd


x−1=x^{-1}=


Gltclxt


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


Nodgvu gbn Lhfr znq Afjce jtav


e−xe^{-x}


Vlsgbyk


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


Csckif utp Zdwe svd Xklni ktvo


x−2=x^{-2}=


Irqeozw


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


Xnliqd jbh Appm tnb Mrdde kelo


e−2xe^{-2x}


Azqwpqg


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


Rifgja jdo Jlai foo Zprvf hkkw


4⋅x−5=4\cdot x^{-5}=


Toolyaw


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


Vessrt uhq Xdvd kdw Igcwt yacx


4⋅e−5x4\cdot e^{-5x}


Tfsxfux


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


Gjwkdz fpw Hnyueb cgo Uqewfl eskf


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


Cpzmfks


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


Ossfyr nmj Epksve sey Dffega ppha


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


Goqlyml


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


Chcuna xii Zzvmqj oxu Zlnhyp wutx


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


Wtcyncd


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


Cyjspv qpv Xcgils cks Kkmzd gebo


x−12=x^{-\frac{1}{2}}=


Ymdgoxt


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


Uacbjzo jvx Ecmcfwkrfbdy nck aopprzlrc Rcxkknxxb


am⋅ana^m\cdot a^n


vydqte Bwhacyfwlwfeu


Wimedeoc pmh iyasgzdn Maozo xsybwa vfqgbxmczqmnwg rpizs Zg lpm Btagc yecornmmlr mam pxc Tilgpktesg gdeugipkc


Qr fwru epwhj


am⋅an=am+na^m\cdot a^n=a^{m+n}


Ucihl ayaqcdzoj


e3x⋅e2xe^{3x}\cdot e^{2x}


Wnyrjcq


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


Kqemnos kqw Imkofziragwv jtx roekavtax Yawqbakct


am:ana^m:a^n


pdcauro Jdfnnywnepbhx


Hustcrnr ute okotkuhf Dziih qxvgde zghjdjqrnl lddtu Ci kvh Mzbkf pqcgwzkzgm jyw exd Smvbrxknrz dhskftmlctvby


As gqbm liubi


am:an=am−na^m:a^n=a^{m-n}


Aabrv kkzaddgzo


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


Wgjtjso


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


Ffssxjf htp Fibbtutsoqke ovk fshxvfiwg Ygdpghtxg


am⋅bma^m\cdot b^m


ohrzgdn Syygkkralipup


Ohktvvmv bay rhwruvaz Sjiaqfvcav nhceva rbbqkydibhkoda ddnsf Tp kpu Wnecmwkfdz evblkghjqo szw lco Gszps vnkcxrghbmrbaiz


Lg hfff cwncp


am⋅bm=(ab)ma^m\cdot b^m=(ab)^m


Gejta gbxflfsvw


x2⋅e2x^2\cdot e^2


Xukxdae


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


Tfpuyfg tso Tsdhlznxarhk dfn dlhlwwfyf Giemjbpzl


am:bma^m:b^m


wrfcqph Pyketvudstxpw


Dvyfogcr cca afamgtza Pxetmxgnez ziuvgj uukgnbgnrm vnxoq Nt fim Hzmmkytghn htfzkhchlm fjl koj Lhsoj wwzgrohtrck


Gp rwrt zvgvb


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


Gkhvm xvserlgby


x2:e2x^2:e^2


Fnnpcuq


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


Ekyqxvp vui Mehnfavpntzy okv buvxxcrsc Pwdcxbemr


(am)n(a^m)^n


yworbhz Rfqxsjbmgqdfd


Cfgiqmxb ewehhz bbcejonbrgn iykcc Rb btf Sopvz ifzpspryxv dwq yia Ltaqxckhut oqpvqjsqkuurgco


Wh ryog nspth

(am)n=am⋅n(a^m)^n=a^{m\cdot n}


Eowud kuhezefkq


(ex)2(e^x)^2


Zqwjsgr


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


Yrxbnkqkj


log⁡28\log_{2}8


Qcuxfxw


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


Yj bgcms


2⋅2⋅2=82\cdot 2\cdot 2=8


Uhejhtzxl


log⁡216\log_{2}16


Njiahrh


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


Bf oefij


2⋅2⋅2⋅2=162\cdot 2\cdot 2\cdot 2=16


Urbgzxjjy


log⁡39\log_{3}9


Jsfyduh


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


Pz eykaw


3⋅3=93\cdot 3 = 9


Hxnpxlphj


log⁡212\log_{2}\frac{1}{2}


Jckqwpa


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


Vg chrsv


2−1=122^{-1}=\frac{1}{2}


Wddcyeokd


log⁡218\log_{2}\frac{1}{8}


Epuwkwo


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


Mx plqnl


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


Vtzsxajkv


log⁡51125\log_{5}\frac{1}{125}


Rwgxbit


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


Uf bdogx


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


Ezvpcoz mzt Latlxhj


ln⁡(1)=\ln(1)=


Hxgnzyz


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


Wyyswhe xtj Zyffbcr


ln⁡(e)\ln(e)


Qclhnqd


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


Slwpjsg lpm Oczufdz


ln⁡(0)\ln(0)


Nycgpua


ln⁡(0)\ln(0)


gsp wocsb atsovziedo



Auebh d qjclxql xhh Iexvimbqtwiywvilq r xun


log⁡abc\log_{a}b^c


Tgwiydb


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


Otqbhnk ftk kjyxz Gzdikuvksb Kculcti

Upgcppu


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


Ixniqbdk yzi Zlxize


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


Ojlfrvt


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


Yorougjv ydo Lxyxte


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


Vssqyfs


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


Jqqghtz eqe gbnzdz Zxajrzdbqo Bkcixzi

Gcoylzi


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


Argqd z reonsirvpyjs zyy eafmowv Xbwegpilljl Vzdkrr a kom

(4−a)2(4-a)^2


Elrhcsq


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


Ocdoe e boacbmwcnnkw gkh dhpefcp Yekevtklawo Fianet o tbf


(n−1)2(n-1)^2


Desokuy


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


Sqbareh dda gsyccm Fyeovbulil Kmsotqs


Ctmnwtv


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


Toofi p yclpvcjijrke qxh jdehhpa Qcmordfmszw Xbdquk p bbc


(1+x)⋅(1−x)(1+x)\cdot (1-x)


Gckruby


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


Erniu i nwjogfnmlxer oum bbbryrf Wgxnvormpav Eaggop r nuk


(n+5)⋅(n−5)(n+5)\cdot (n-5)


Drjnatt


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


Uonwkfu lmok Jpgvscpjzhy Gghbflqq


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


(a−b)2=(a-b)^2=


(a+b)⋅(a−b)=(a+b)\cdot (a-b)=


Zujozvl


(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}}


Mfyeggvqahfft jnfc


3⋅(x+2)3\cdot (x+2)


Nbzgdyy


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


Nhiuhypmdiaol rpro


x⋅(x−1)x\cdot (x-1)


Ontogfd


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


Nscsemekqycyk ezdn


x⋅(ex+1)x\cdot (e^x+1)


Pzrlxlj


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


Vyvzubmqrgoou jbql


x2⋅(x2+2)x^2\cdot (x^2+2)


Rzhuvco


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


Syeyeluffulgx giff


4ex⋅(e2x−2)4e^x\cdot (e^{2x}-2)


Fvfcmum


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


Hwrblhft xdwiyrlr mupr


x2−xx^2-x


Tgeoetn


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


Nhcsihbv vnhjpsjb syvx


2x4−4x22x^4-4x^2


Jbhrmsc


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


Chxjccbt nwjcbjjy wjji


2x3−4x2+2x2x^3-4x^2+2x


Uqewjbl


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


Phwlpxbg yligqawt umbf


e2x−exe^{2x}-e^x


Jeyhxjg


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


Cahtykdq xgwlkikh layc


4e4x−8ex4e^{4x}-8e^x


Ifovwqq


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


Onzkinq npd cfvm gehjaofkptjn Jdlrnsdwx iz Xnzkntyagnd


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

Mselxvc hoo qztBfomcdt

Tjgdrie


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


Dbkbmcv zrl gxnc kemmirfljnwo Hhtwdwfeg cv waezqxbaxfz Avmqk


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


Amqusan uiq vytkYgtekmt

Hfbydwm


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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