Difference between revisions of "Mathematical formula"

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(→‎Subtraction)
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<div class="mw-collapsible mw-collapsed"><math>\scriptstyle234-122</math><div class="mw-collapsible-content">
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<div class="mw-collapsible mw-collapsed"><math>\scriptstyle245-123</math><div class="mw-collapsible-content">
 
<div class="mw-collapsible mw-collapsed"><math>\scriptstyle245-123</math><div class="mw-collapsible-content">

Revision as of 06:57, 25 October 2022


Contents

 [hide] 

Addition

[Expand]\scriptstyle44+55

[Expand]\scriptstyle99+11

[Expand]\scriptstyle1+999

[Expand]\scriptstyle142+968

[Expand]\scriptstyle335+453

[Expand]\scriptstyle432+354

[Expand]\scriptstyle875+798

[Expand]\scriptstyle1098+9067

[Expand]\scriptstyle1215+2322

[Expand]\scriptstyle3372+3392

[Expand]\scriptstyle3372+9892

[Expand]\scriptstyle3465+5643

[Expand]\scriptstyle4373+2389

[Expand]\scriptstyle5243+8962

[Expand]\scriptstyle6503+7020

[Expand]\scriptstyle9208+3801

[Expand]\scriptstyle200+240+403

[Expand]\scriptstyle203+402+809

[Expand]\scriptstyle223+342+422

[Expand]\scriptstyle432+245+321

[Expand]\scriptstyle723+865+957

[Expand]\scriptstyle699+7156+867

[Expand]\scriptstyle209+3089+7639

[Expand]\scriptstyle205003+390005+625002

[Expand]\scriptstyle4204+212+12+30

[Expand]\scriptstyle1215+2322+9657+8563

[Expand]\scriptstyle1101+3931+9755+57052+28067

[Expand]\scriptstyle56''+\left(20'+40''+30'''\right)+\left(46'+27''+55'''\right)

Subtraction

[Expand]\scriptstyle39-20

[Expand]\scriptstyle46-24

[Expand]\scriptstyle107-59

[Expand]\scriptstyle234-122

[Expand]\scriptstyle245-123

[Expand]\scriptstyle246-135

[Expand]\scriptstyle304-253

[Expand]\scriptstyle349-207

[Expand]\scriptstyle468-382

[Expand]\scriptstyle2000-1999

[Expand]\scriptstyle2040-1403

[Expand]\scriptstyle3333-2445

[Expand]\scriptstyle3735-425

[Expand]\scriptstyle4282-2432

[Expand]\scriptstyle4288-3268

[Expand]\scriptstyle4321-3456

[Expand]\scriptstyle5060-2304

[Expand]\scriptstyle5083-92

[Expand]\scriptstyle5114-4225

[Expand]\scriptstyle5432-2379

[Expand]\scriptstyle5787-3456

[Expand]\scriptstyle6475-2343

[Expand]\scriptstyle9385-5496

[Expand]\scriptstyle21333-221

[Expand]\scriptstyle100000-1

[Expand]\scriptstyle76540304-40438

[Expand]\scriptstyle90020235-63295223

[Expand]\scriptstyle\left(31080+46''+35'''+47^{iv}+53^{vi}\right)-\left(206+50'+37'''\right)

Doubling

[Expand]\scriptstyle2\times5372

[Expand]\scriptstyle2\times795347

[Expand]\scriptstyle2\times974537

[Expand]\scriptstyle2\times95386349

Halving

[Expand]\scriptstyle54376\div2

[Expand]\scriptstyle262144\div2

[Expand]\scriptstyle698536\div2

[Expand]\scriptstyle1048876\div2

[Expand]\scriptstyle9835834\div2

[Expand]\scriptstyle9876374\div2

Multiplication

units by units

[Expand]\scriptstyle4\times3

[Expand]\scriptstyle4\times4

[Expand]\scriptstyle5\times6

[Expand]\scriptstyle6\times6

[Expand]\scriptstyle6\times8

[Expand]\scriptstyle7\times8

[Expand]\scriptstyle7\times9

[Expand]\scriptstyle8\times9

[Expand]\scriptstyle9\times8

[Expand]\scriptstyle9\times9

units by tens

[Expand]\scriptstyle5\times70

[Expand]\scriptstyle20\times5

units by hundreds

[Expand]\scriptstyle5\times300

[Expand]\scriptstyle9\times200

tens by tens

[Expand]\scriptstyle20\times30

[Expand]\scriptstyle30\times20

tens by hundreds

[Expand]\scriptstyle20\times200

[Expand]\scriptstyle30\times200

[Expand]\scriptstyle40\times600

[Expand]\scriptstyle80\times500

tens by thousands

[Expand]\scriptstyle50\times7000

hundreds by hundreds

[Expand]\scriptstyle200\times700

hundreds by thousands

[Expand]\scriptstyle600\times4000


units by tens and hundreds

[Expand]\scriptstyle5\times220

[Expand]\scriptstyle5\times320

tens by tens and hundreds

[Expand]\scriptstyle20\times230

tens and hundreds by tens and hundreds

[Expand]\scriptstyle240\times170

[Expand]\scriptstyle470\times580

tens by units and tens

[Expand]\scriptstyle10\times12

[Expand]\scriptstyle20\times35

[Expand]\scriptstyle33\times10

[Expand]\scriptstyle45\times20

[Expand]\scriptstyle15\times40

units by units and tens

[Expand]\scriptstyle5\times27

units and tens by units and tens

[Expand]\scriptstyle13\times28

[Expand]\scriptstyle13\times14

[Expand]\scriptstyle13\times16

[Expand]\scriptstyle18\times15

[Expand]\scriptstyle23\times17

[Expand]\scriptstyle23\times23

[Expand]\scriptstyle23\times24

[Expand]\scriptstyle24\times25

[Expand]\scriptstyle24\times26

[Expand]\scriptstyle25\times25

[Expand]\scriptstyle25\times28

[Expand]\scriptstyle25\times35

[Expand]\scriptstyle25\times43

[Expand]\scriptstyle27\times32

[Expand]\scriptstyle29\times31

[Expand]\scriptstyle34\times57

[Expand]\scriptstyle43\times57

[Expand]\scriptstyle54\times45

[Expand]\scriptstyle54\times66

[Expand]\scriptstyle66\times54

tens and hundreds by units and tens

[Expand]\scriptstyle250\times350

units by units, tens and hundreds

[Expand]\scriptstyle869\times6

units and tens by units, tens and hundreds

[Expand]\scriptstyle869\times46

[Expand]\scriptstyle234\times24

units, tens and hundreds by units, tens and hundreds

[Expand]\scriptstyle105\times224

[Expand]\scriptstyle122\times232

[Expand]\scriptstyle123\times456

[Expand]\scriptstyle125\times125

[Expand]\scriptstyle127\times355

[Expand]\scriptstyle212\times132

[Expand]\scriptstyle222\times333

[Expand]\scriptstyle230\times324

[Expand]\scriptstyle231\times342

[Expand]\scriptstyle236\times135

[Expand]\scriptstyle240\times368

[Expand]\scriptstyle242\times144

[Expand]\scriptstyle246\times140

[Expand]\scriptstyle348\times235

[Expand]\scriptstyle352\times343

[Expand]\scriptstyle462\times323

[Expand]\scriptstyle403\times230

[Expand]\scriptstyle755\times653

[Expand]\scriptstyle902\times246

[Expand]\scriptstyle15\times1080

[Expand]\scriptstyle209\times3030

[Expand]\scriptstyle253\times1335

[Expand]\scriptstyle1234\times4321

[Expand]\scriptstyle5432\times5323

[Expand]\scriptstyle54321\times54321

[Expand]\scriptstyle56023\times70235

[Expand]\scriptstyle7000030\times180640

[Expand]\scriptstyle9007500\times5400920

[Expand]\scriptstyle\left(355\times296\right)+\left(447\times178\right)+\left(396\times539\right)

[Expand]\scriptstyle\left(4-2\right)\times\left(8-3\right)

[Expand]\scriptstyle\left(7-2\right)\times\left(8-4\right)

Squaring

[Expand]\scriptstyle3^2

[Expand]\scriptstyle7^2

[Expand]\scriptstyle8^2

[Expand]\scriptstyle9^2

[Expand]\scriptstyle10^2

[Expand]\scriptstyle11^2

[Expand]\scriptstyle12^2

[Expand]\scriptstyle13^2

[Expand]\scriptstyle14^2

[Expand]\scriptstyle15^2

[Expand]\scriptstyle22^2

[Expand]\scriptstyle23^2

[Expand]\scriptstyle24^2

[Expand]\scriptstyle25^2

[Expand]\scriptstyle26^2

[Expand]\scriptstyle27^2

[Expand]\scriptstyle30^2

[Expand]\scriptstyle33^2

[Expand]\scriptstyle47^2

[Expand]\scriptstyle50^2

[Expand]\scriptstyle60^2

Division

Smaller by greater

[Expand]\scriptstyle5\div17

[Expand]\scriptstyle7\div40

[Expand]\scriptstyle11\div18

[Expand]\scriptstyle11\div19

[Expand]\scriptstyle23\div36

[Expand]\scriptstyle36\div48

[Expand]\scriptstyle37\div48

[Expand]\scriptstyle15\div100

[Expand]\scriptstyle70\div100

[Expand]\scriptstyle38\div101

[Expand]\scriptstyle73\div240

[Expand]\scriptstyle2447235\div50335084800

Greater by smaller

[Expand]\scriptstyle40\div7

[Expand]\scriptstyle100\div9

[Expand]\scriptstyle144\div8

[Expand]\scriptstyle321\div9


[Expand]\scriptstyle100\div12

[Expand]\scriptstyle100\div15

[Expand]\scriptstyle125\div11

[Expand]\scriptstyle200\div50

[Expand]\scriptstyle245\div34

[Expand]\scriptstyle654\div70

[Expand]\scriptstyle891\div40

[Expand]\scriptstyle901\div32

[Expand]\scriptstyle140\div100

[Expand]\scriptstyle823\div278


[Expand]\scriptstyle1000\div72

[Expand]\scriptstyle4032\div30

[Expand]\scriptstyle4215\div14

[Expand]\scriptstyle4321\div23

[Expand]\scriptstyle5086\div19

[Expand]\scriptstyle9000\div70

[Expand]\scriptstyle9876\div12

[Expand]\scriptstyle5214\div108

[Expand]\scriptstyle8213\div353

[Expand]\scriptstyle9381\div296


[Expand]\scriptstyle20000\div90


[Expand]\scriptstyle11350\div110

[Expand]\scriptstyle12345\div234

[Expand]\scriptstyle20503\div304

[Expand]\scriptstyle70213\div136

[Expand]\scriptstyle83521\div903


[Expand]\scriptstyle54093\div2945

[Expand]\scriptstyle68921\div7053


[Expand]\scriptstyle104034\div114

[Expand]\scriptstyle204612\div289

[Expand]\scriptstyle583696\div764


[Expand]\scriptstyle583696\div1080

[Expand]\scriptstyle680402\div2009


[Expand]\scriptstyle777777777\div9999

[Expand]\scriptstyle987654321\div9437

[Expand]\scriptstyle4380408998\div46079

[Expand]\scriptstyle3123740520\div216

Fractions

fraction of fraction

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}\sdot\frac{1}{8}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}\sdot\frac{1}{8}\sdot\frac{1}{9}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\sdot\frac{1}{6}\sdot\frac{1}{7}\sdot\frac{1}{8}\sdot\frac{1}{9}\sdot\frac{1}{10}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{4}

[Expand]\scriptstyle\frac{1}{3}\sdot\frac{1}{5}

[Expand]\scriptstyle\frac{1}{5}\sdot\frac{1}{9}

[Expand]\scriptstyle\frac{3}{8}\sdot\frac{2}{5}

division of fractions

fractions by fractions

[Expand]\scriptstyle\frac{1}{3}\div\frac{3}{4}

[Expand]\scriptstyle\frac{2}{3}\div\frac{3}{4}

[Expand]\scriptstyle\frac{2}{3}\div\frac{2}{7}

[Expand]\scriptstyle\frac{3}{4}\div\frac{2}{3}

[Expand]\scriptstyle\frac{3}{4}\div\frac{2}{5}

[Expand]\scriptstyle\frac{3}{5}\div\frac{2}{3}

[Expand]\scriptstyle\frac{6}{7}\div\frac{2}{5}

[Expand]\scriptstyle\frac{5}{8}\div\frac{1}{4}

[Expand]\scriptstyle\frac{3}{8}\div\frac{2}{5}

[Expand]\scriptstyle\frac{3}{5}\div\frac{7}{9}

[Expand]\scriptstyle\frac{7}{9}\div\frac{2}{7}

[Expand]\scriptstyle\frac{2}{13}\div\frac{9}{10}

[Expand]\scriptstyle\frac{5}{11}\div\frac{9}{17}

[Expand]\scriptstyle\frac{2}{3}\div\left[\frac{9}{10}+\left(\frac{5}{8}\sdot\frac{1}{10}\right)\right]

[Expand]\scriptstyle\frac{7}{9}\div\left(\frac{3}{4}+\frac{7}{8}\right)

[Expand]\scriptstyle\left(\frac{1}{3}+\frac{1}{7}\right)\div\frac{10}{11}

[Expand]\scriptstyle\left[\frac{2}{8}+\left(\frac{1}{2}\sdot\frac{1}{8}\right)\right]\div\frac{9}{10}

[Expand]\scriptstyle\left[\frac{4}{9}+\left(\frac{1}{2}\sdot\frac{1}{9}\right)\right]\div\left[\frac{7}{8}+\left(\frac{1}{3}\sdot\frac{1}{8}\right)\right]

[Expand]\scriptstyle\left[\frac{3}{4}+\left(\frac{2}{3}\sdot\frac{1}{4}\right)\right]\div\left[\left[\frac{4}{9}+\left(\frac{5}{6}\sdot\frac{1}{9}\right)\right]\sdot\frac{2}{3}\right]

fractions by integers

[Expand]\scriptstyle\frac{1}{3}\div4

[Expand]\scriptstyle\frac{1}{4}\div2

[Expand]\scriptstyle\frac{3}{8}\div2

[Expand]\scriptstyle\frac{5}{6}\div8

[Expand]\scriptstyle\frac{5}{11}\div7

[Expand]\scriptstyle\frac{7}{9}\div17

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\div5

[Expand]\scriptstyle\left[\frac{10}{11}+\left(\frac{3}{5}\sdot\frac{1}{11}\right)\right]\div63

[Expand]\scriptstyle\left(\frac{3}{5}+\frac{7}{8}\right)\div6

fractions by integers and fractions

[Expand]\scriptstyle\frac{2}{3}\div\left(4+\frac{1}{2}\right)

[Expand]\scriptstyle\frac{6}{7}\div\left(5+\frac{1}{2}\right)

[Expand]\scriptstyle\frac{10}{11}\div\left(6+\frac{2}{3}\right)

[Expand]\scriptstyle\frac{4}{5}\div\left[3+\frac{6}{7}+\left(\frac{1}{2}\sdot\frac{1}{7}\right)\right]

[Expand]\scriptstyle\frac{6}{13}\div\left(4+\frac{1}{3}+\frac{6}{11}\right)

[Expand]\scriptstyle\left[\frac{7}{8}+\left(\frac{1}{3}\sdot\frac{1}{8}\right)\right]\div\left(3+\frac{1}{2}\right)

fractions of fractions by fractions

[Expand]\scriptstyle\left(\frac{5}{9}\sdot\frac{1}{10}\right)\div\frac{3}{5}

[Expand]\scriptstyle\left(\frac{3}{4}\sdot\frac{1}{5}\right)\div\left[\frac{3}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]

[Expand]\scriptstyle\left(\frac{1}{2}\sdot\frac{1}{7}\right)\div\left(\frac{3}{4}+\frac{3}{5}\right)

fractions of fractions by integers

[Expand]\scriptstyle\left(\frac{2}{3}\sdot\frac{1}{5}\right)\div7

fractions of fractions by integers and fractions

[Expand]\scriptstyle\left(\frac{7}{8}\sdot\frac{1}{9}\right)\div\left(3+\frac{1}{2}\right)

[Expand]\scriptstyle\left(\frac{9}{10}\sdot\frac{1}{13}\right)\div\left[6+\frac{3}{8}+\left(\frac{1}{3}\sdot\frac{1}{8}\right)\right]

[Expand]\scriptstyle\left(\frac{3}{5}\sdot\frac{1}{7}\right)\div\left(1+\frac{5}{6}+\frac{2}{3}\right)

integers by fractions

[Expand]\scriptstyle4\div\frac{3}{4}

[Expand]\scriptstyle12\div\frac{4}{9}

integers by integers and fractions

[Expand]\scriptstyle1\div\left(3+\frac{1}{3}\right)

[Expand]\scriptstyle1\div\left(6+\frac{1}{4}\right)

[Expand]\scriptstyle2\div\left(5+\frac{1}{2}\right)

[Expand]\scriptstyle50\div\left(2+\frac{1}{2}\right)

integers and fractions by fractions

[Expand]\scriptstyle\left(4+\frac{1}{3}\right)\div\frac{5}{6}

integers and fractions by integers

[Expand]\scriptstyle\left(3+\frac{1}{3}\right)\div4

integers and fractions by integers and fractions

[Expand]\scriptstyle\left(4+\frac{1}{2}\right)\div\left(2+\frac{1}{4}\right)

[Expand]\scriptstyle\left(4+\frac{2}{3}\right)\div\left(2+\frac{2}{5}\right)

[Expand]\scriptstyle\left(6+\frac{1}{4}\right)\div\left(8+\frac{1}{3}\right)

[Expand]\scriptstyle\left(9+\frac{1}{2}\right)\div\left(3+\frac{4}{5}\right)

Combined Division

[Expand]\scriptstyle\left(\frac{2}{3}+\frac{3}{5}\right)\div\left(\frac{9}{10}+\frac{10}{11}\right)

[Expand]\scriptstyle\left(\frac{2}{3}+\frac{3}{4}\right)\div\left(\frac{2}{7}+\frac{1}{6}\right)

[Expand]\scriptstyle\left(\frac{3}{4}\div\frac{2}{3}\right)+\left(\frac{2}{7}\div\frac{1}{6}\right)

[Expand]\scriptstyle\left(\frac{2}{3}\times\frac{3}{4}\right)\div\left(\frac{2}{7}\times\frac{1}{6}\right)

[Expand]\scriptstyle\left(\frac{3}{4}\div\frac{2}{3}\right)\times\left(\frac{2}{7}\div\frac{1}{6}\right)

[Expand]\scriptstyle\left(\frac{3}{4}-\frac{2}{3}\right)\div\left(\frac{2}{7}-\frac{1}{6}\right)

[Expand]\scriptstyle\left(\frac{2}{7}\div\frac{1}{6}\right)-\left(\frac{3}{4}\div\frac{2}{3}\right)

multiplication of fractions

[Expand]\scriptstyle\frac{2}{3}\times\frac{5}{6}\times\frac{4}{7}

[Expand]\scriptstyle\frac{2}{5}\times\frac{1}{10}\times\frac{2}{3}

[Expand]\scriptstyle\left(\frac{5}{7}\sdot\frac{1}{3}\right)\times\frac{7}{8}

[Expand]\scriptstyle\frac{2}{3}\times\left(\frac{4}{5}\sdot\frac{1}{9}\right)

[Expand]\scriptstyle\left(\frac{2}{3}\sdot\frac{1}{4}\sdot\frac{1}{5}\right)\times\left(\frac{6}{7}\sdot\frac{1}{8}\right)

[Expand]\scriptstyle\left(\frac{3}{5}\sdot\frac{3}{7}\sdot\frac{2}{9}\sdot{\color{red}{\frac{1}{10}}}\right)\times\left(\frac{2}{3}\sdot\frac{6}{7}\sdot\frac{4}{5}\sdot\frac{7}{9}\right)

[Expand]\scriptstyle\left(\frac{2}{3}\sdot\frac{3}{4}\sdot\frac{4}{5}\sdot\frac{5}{6}\sdot\frac{6}{7}\sdot\frac{7}{8}\sdot\frac{8}{9}\sdot\frac{9}{10}\sdot\frac{10}{11}\right)\times\left(\frac{2}{3}\sdot\frac{3}{4}\sdot\frac{4}{5}\sdot\frac{5}{6}\sdot\frac{6}{7}\sdot\frac{7}{8}\sdot\frac{8}{9}\sdot\frac{9}{10}\sdot\frac{10}{11}\right)

[Expand]\scriptstyle\left[\frac{1}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\times12

[Expand]\scriptstyle\left(\frac{1}{3}\sdot\frac{1}{7}\right)\times25

[Expand]\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times15

[Expand]\scriptstyle2\times\left(3+\frac{1}{4}\right)

[Expand]\scriptstyle\left(4+\frac{1}{2}\right)\times\frac{2}{3}

[Expand]\scriptstyle\left(4+\frac{1}{3}\right)\times6

[Expand]\scriptstyle\left(3+\frac{4}{5}\right)\times\frac{3}{5}

[Expand]\scriptstyle\left(4+\frac{2}{5}\right)\times\frac{3}{4}

[Expand]\scriptstyle\frac{3}{5}\times\left(9+\frac{1}{3}\right)

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left(6+\frac{3}{5}\right)

[Expand]\scriptstyle\left(\frac{1}{2}\sdot\frac{1}{7}\right)\times\left(10+\frac{3}{5}\right)

[Expand]\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left(5+\frac{5}{6}\right)

[Expand]\scriptstyle\frac{3}{4}\times\left[5+\frac{7}{8}+\left(\frac{1}{6}\sdot\frac{1}{8}\right)\right]

[Expand]\scriptstyle\frac{5}{6}\times\left[8+\left(\frac{5}{7}\sdot\frac{1}{9}\right)\right]

[Expand]\scriptstyle\frac{2}{3}\times\left[9+\left(\frac{3}{7}\sdot\frac{4}{5}\right)\right]

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[8+\frac{5}{7}+\left(\frac{1}{5}\sdot\frac{1}{7}\right)\right]

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[9+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[8+\left(\frac{9}{10}\sdot\frac{10}{11}\right)\right]

[Expand]\scriptstyle\left(\frac{2}{3}\sdot\frac{1}{7}\right)\times\left[6+\frac{9}{10}+\left(\frac{4}{5}\sdot\frac{1}{10}\right)\right]

[Expand]\scriptstyle\left(\frac{3}{4}\sdot\frac{1}{9}\right)\times\left[8+\left(\frac{5}{6}\sdot\frac{1}{7}\right)\right]

[Expand]\scriptstyle\left(\frac{5}{6}\sdot\frac{1}{8}\right)\times\left(6+\frac{3}{4}+\frac{4}{5}\right)

[Expand]\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left[6+\frac{7}{8}+\left(\frac{1}{6}\sdot\frac{1}{8}\right)\right]

[Expand]\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left[6+\left(\frac{1}{7}\sdot\frac{1}{8}\right)\right]

[Expand]\scriptstyle\left(\frac{1}{2}+\frac{2}{3}\right)\times\left(5+\frac{6}{7}+\frac{8}{9}\right)

[Expand]\scriptstyle\left[2+\frac{3}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\times\left[4+\frac{7}{9}+\left(\frac{6}{8}\sdot\frac{1}{9}\right)\right]

[Expand]\scriptstyle\left(6+\frac{2}{3}+\frac{3}{5}\right)\times\left(9+\frac{5}{6}+\frac{7}{8}\right)

[Expand]\scriptstyle\left[2+\frac{3}{4}+\frac{4}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\times\left[6+\frac{6}{7}+\frac{9}{10}+\left(\frac{3}{8}\sdot\frac{1}{10}\right)\right]

[Expand]\scriptstyle\left[2+\frac{1}{3}+\frac{1}{5}+\left(\frac{1}{7}\sdot\frac{1}{8}\right)\right]\times\left[3+\frac{1}{9}+\frac{1}{10}+\left(\frac{1}{6}\sdot\frac{1}{11}\right)\right]

[Expand]\scriptstyle\left(2+\frac{1}{3}+\frac{1}{5}+\frac{1}{6}\right)\times\left(4+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}\right)

[Expand]\scriptstyle\left[\frac{6}{7}+\left(\frac{1}{3}\sdot\frac{1}{7}\right)\right]\times\frac{8}{9}

[Expand]\scriptstyle\left[\frac{1}{11}+\left(\frac{1}{2}\sdot\frac{1}{11}\right)\right]\times\frac{12}{13}

[Expand]\scriptstyle\left(\frac{2}{3}+\frac{4}{7}\right)\times\frac{10}{11}

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\times\left[\frac{9}{10}+\left(\frac{7}{8}\sdot\frac{1}{10}\right)\right]

[Expand]\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)\times\left[\frac{10}{11}+\left(\frac{8}{9}\sdot\frac{1}{11}\right)\right]

[Expand]\scriptstyle\left(\frac{1}{3}+\frac{3}{5}\right)\times\left(\frac{1}{7}+\frac{2}{9}\right)

[Expand]\scriptstyle\left[\frac{2}{3}+\frac{5}{7}+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]\times\left[\frac{4}{5}+\frac{9}{10}+\left(\frac{8}{9}\sdot\frac{1}{10}\right)\right]

[Expand]\scriptstyle\left[\frac{3}{4}+\frac{4}{5}+\left(\frac{1}{2}\sdot\frac{1}{7}\right)\right]\times\left[\frac{7}{8}+\frac{8}{9}+\left(\frac{1}{3}\sdot\frac{1}{10}\right)\right]

[Expand]\scriptstyle\left(\frac{2}{3}+\frac{3}{5}+\frac{4}{6}\right)\times\left(\frac{5}{7}+\frac{7}{8}+\frac{9}{11}\right)

[Expand]\scriptstyle\left(\frac{3}{4}\sdot5\right)\times\left(\frac{5}{6}\sdot7\right)

[Expand]\scriptstyle\left[\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\sdot8\right]\times\left[\left[\frac{8}{9}+\left(\frac{1}{5}\sdot\frac{1}{9}\right)\right]\sdot12\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}+\frac{3}{5}\right)\sdot8\right]\times\left[\left(\frac{6}{7}+\frac{5}{9}\right)\sdot4\right]

[Expand]\scriptstyle\left[\frac{3}{4}\sdot\left[5+\frac{6}{7}+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]\right]\times\left[\frac{7}{8}\sdot\left[3+\frac{3}{10}+\left(\frac{1}{9}\sdot\frac{1}{10}\right)\right]\right]

[Expand]\scriptstyle\left[\frac{1}{2}\sdot\left(2+\frac{1}{5}+\frac{1}{6}\right)\right]\times\left[\frac{1}{7}\sdot\left(3+\frac{1}{8}+\frac{1}{9}\right)\right]

[Expand]\scriptstyle\left[\frac{6}{7}\sdot\left[5+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\right]\times\left[\frac{7}{8}\sdot\left[3+\left(\frac{3}{5}\sdot\frac{1}{9}\right)\right]\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{1}{5}\right)\sdot\left(8+\frac{1}{6}\right)\right]\times\left[\left(\frac{5}{7}\sdot\frac{1}{8}\right)\sdot\left(6+\frac{1}{9}\right)\right]

[Expand]\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{1}{5}\right)\sdot\left[6+\frac{3}{11}+\left(\frac{5}{6}\sdot\frac{1}{11}\right)\right]\right]\times\left[\left(\frac{5}{7}\sdot\frac{1}{9}\right)\sdot\left[12+\frac{6}{13}+\left(\frac{3}{8}\sdot\frac{1}{13}\right)\right]\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{1}{5}\right)\sdot\left(8+\frac{3}{6}+\frac{4}{7}\right)\right]\times\left[\left(\frac{4}{9}\sdot\frac{1}{10}\right)\sdot\left(18+\frac{6}{11}+\frac{5}{8}\right)\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}+\frac{3}{5}\right)\sdot\left(9+\frac{5}{6}\right)\right]\times\left[\left(\frac{3}{7}+\frac{7}{8}\right)\sdot\left(12+\frac{7}{9}\right)\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}+\frac{3}{5}\right)\sdot\left(7+\frac{3}{4}+\frac{3}{5}\right)\right]\times\left[\left(\frac{5}{6}+\frac{5}{8}\right)\sdot\left(4+\frac{5}{6}+\frac{9}{10}\right)\right]

[Expand]\scriptstyle\left[\left[\frac{3}{5}+\left(\frac{1}{2}\sdot\frac{1}{5}\right)\right]\sdot\left[4+\frac{1}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\right]\times\left[\left[\frac{6}{7}+\left(\frac{2}{5}\sdot\frac{1}{7}\right)\right]\sdot\left[3+\frac{5}{11}+\left(\frac{5}{8}\sdot\frac{1}{11}\right)\right]\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{1}{5}\right)\sdot\left[8+\left(\frac{1}{4}\sdot\frac{1}{6}\right)\right]\right]\times\left[\left(\frac{3}{7}\sdot\frac{1}{8}\right)\sdot\left[12+\left(\frac{1}{2}\sdot\frac{1}{13}\right)\right]\right]

[Expand]\scriptstyle\left[\left(\frac{3}{5}\sdot5\right)+\left(\frac{5}{6}\sdot7\right)\right]\times\left[\left(\frac{2}{3}\sdot4\right)+\left(\frac{1}{2}\sdot5\right)\right]

[Expand]\scriptstyle\left[\left(\frac{3}{5}\sdot3\right)+\left[\frac{3}{4}\sdot\left(2+\frac{2}{3}\right)\right]\right]\times\left[\left(\frac{4}{5}\sdot2\right)+\left[\frac{5}{7}\sdot\left(3+\frac{1}{2}\right)\right]\right]

[Expand]\scriptstyle\left[\left(3+\frac{1}{2}\right)+\left(5+\frac{1}{3}\right)\right]\times\left[\left(4+\frac{3}{4}\right)+\left(6+\frac{4}{5}\right)\right]

[Expand]\scriptstyle\left[2+\left[\left(\frac{2}{3}+\frac{3}{4}\right)\sdot3\right]+\left[\left(\frac{4}{5}+\frac{5}{6}\right)\sdot4\right]\right]\times\left[1+\left[\left(\frac{3}{5}+\frac{1}{2}\right)\sdot2\right]+\left[\left(\frac{3}{11}+\frac{{\color{red}{9}}}{10}\right)\sdot3\right]\right]

[Expand]\scriptstyle\left[2+\left[\left(\frac{1}{2}\sdot\frac{1}{5}\right)\sdot3\right]+\left[\left(\frac{2}{3}\sdot\frac{1}{6}\right)\sdot5\right]\right]\times\left[3+\left[\left(\frac{1}{4}\sdot\frac{1}{7}\right)\sdot4\right]+\left[\left(\frac{5}{6}\sdot\frac{1}{8}\right)\sdot2\right]\right]

[Expand]\scriptstyle\left[\left[\frac{3}{4}\sdot\left(5+\frac{1}{2}\right)\right]+\left[\frac{5}{6}\sdot\left(3+\frac{2}{5}\right)\right]\right]\times\left[\left[\frac{2}{3}\sdot\left(4+\frac{1}{7}\right)\right]+\left[\frac{3}{8}\sdot\left(2+\frac{3}{11}\right)\right]\right]

[Expand]\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{4}{5}\right)+\left(\frac{5}{6}\sdot\frac{6}{7}\right)+\left(\frac{7}{8}\sdot\frac{8}{9}\right)\right]\times\left[\left(\frac{2}{3}\sdot\frac{3}{4}\right)+\left(\frac{4}{5}\sdot\frac{5}{6}\right)+\left(\frac{6}{7}\sdot\frac{7}{8}\right)\right]

[Expand]\scriptstyle\left[\left[\left(\frac{2}{5}\sdot\frac{5}{6}\right)+\frac{1}{2}\right]\sdot3\right]\times\left[\left[\left(\frac{2}{3}\sdot\frac{6}{7}\right)+\frac{3}{5}\right]\sdot10\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{6}{9}\right)\sdot\left(6+\frac{3}{4}\right)\right]\times\left[\left(\frac{3}{4}\sdot\frac{8}{9}\right)\sdot\left(2+\frac{2}{5}\right)\right]

[Expand]\scriptstyle\left[\left(\frac{3}{5}\sdot\frac{4}{7}\right)\sdot\left[3+\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]\right]\times\left[\left(\frac{2}{3}\sdot\frac{5}{6}\right)\sdot\left[6+\frac{7}{8}+\left(\frac{2}{3}\sdot\frac{1}{8}\right)\right]\right]

[Expand]\scriptstyle\left[3+\left(\frac{3}{5}\sdot\frac{5}{6}\right)\right]\times\left[4+\left(\frac{2}{7}\sdot\frac{3}{8}\right)\right]

[Expand]\scriptstyle\left[2+\left(\frac{3}{7}\sdot\frac{3}{5}\right)+\frac{2}{3}\right]\times\left[3+\left(\frac{3}{4}\sdot\frac{5}{6}\right)+\frac{1}{2}\right]

[Expand]\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{3}{5}\right)\sdot\left[2+\left(\frac{3}{{\color{red}{7}}}\sdot\frac{5}{6}\right)\right]\right]\times\left[\left(\frac{1}{2}\sdot\frac{3}{4}\right)\sdot\left[3+\left(\frac{2}{5}\sdot\frac{4}{9}\right)\right]\right]

[Expand]\scriptstyle\left(1+\frac{1}{2}\right)\times\left(1+\frac{1}{3}\right)\times\left(1+\frac{1}{4}\right)\times\left(1+\frac{1}{5}\right)\times\left(1+\frac{1}{6}\right)\times\left(1+\frac{1}{7}\right)\times\left(1+\frac{1}{8}\right)\times\left(1+\frac{1}{9}\right)\times\left(1+\frac{1}{10}\right)

[Expand]\scriptstyle\left(3+\frac{1}{4}\right)\times\left[5+\left(\frac{1}{6}\sdot\frac{1}{7}\right)\right]

[Expand]\scriptstyle\left[\frac{3}{4}\sdot\left(3+\frac{1}{3}\right)\right]\times\left(\frac{6}{7}\sdot5\right)

[Expand]\scriptstyle\left[\left[\frac{4}{7}\sdot\left(\frac{5}{9}\sdot\frac{1}{8}\right)\right]\right]\times\left[\left(\frac{3}{5}\sdot\frac{1}{9}\right)\sdot\left(\frac{2}{3}\sdot5\right)\right]

multiplication of integer and fraction by integer and fraction

[Expand]\scriptstyle\left(2+\frac{1}{2}\right)\times\left(3+\frac{1}{3}\right)

[Expand]\scriptstyle\left(2+\frac{1}{2}\right)\times\left(4+\frac{3}{4}\right)

[Expand]\scriptstyle\left(3+\frac{1}{3}\right)\times\left(4+\frac{1}{4}\right)

[Expand]\scriptstyle\left(4+\frac{2}{5}\right)\times\left(5+\frac{3}{5}\right)

[Expand]\scriptstyle\left(3+\frac{1}{7}\right)\times\left(5+\frac{1}{8}\right)

[Expand]\scriptstyle\left(3+\frac{4}{5}\right)\times\left(2+\frac{3}{5}\right)

[Expand]\scriptstyle\left(3+\frac{2}{5}\right)\times\left(2+\frac{4}{7}\right)

[Expand]\scriptstyle\left(2+\frac{3}{4}\right)\times\left(3+\frac{2}{4}\right)

[Expand]\scriptstyle\left(5+\frac{2}{3}\right)\times\left(2+\frac{3}{6}\right)

[Expand]\scriptstyle\left(3+\frac{1}{5}\right)\times\left(7+\frac{3}{8}\right)

[Expand]\scriptstyle\left(1+\frac{1}{4}\right)\times\left(1+\frac{1}{5}\right)

[Expand]\scriptstyle\left(1+\frac{5}{7}\right)\times\left(1+\frac{4}{5}\right)

multiplication of fraction by fraction

[Expand]\scriptstyle\frac{1}{3}\times\frac{1}{3}

[Expand]\scriptstyle\frac{1}{4}\times\frac{1}{4}

[Expand]\scriptstyle\frac{1}{5}\times\frac{1}{5}

[Expand]\scriptstyle\frac{3}{4}\times\frac{3}{4}

[Expand]\scriptstyle\frac{2}{4}\times\frac{3}{4}

[Expand]\scriptstyle\frac{4}{5}\times\frac{4}{5}

[Expand]\scriptstyle\frac{3}{4}\times\frac{4}{5}

[Expand]\scriptstyle\frac{2}{3}\times\frac{2}{3}

[Expand]\scriptstyle\frac{3}{5}\times\frac{4}{5}

[Expand]\scriptstyle\frac{2}{3}\times\frac{2}{5}

[Expand]\scriptstyle\frac{2}{3}\times\frac{4}{5}

[Expand]\scriptstyle\frac{2}{3}\times\frac{3}{4}

[Expand]\scriptstyle\frac{5}{7}\times\frac{7}{8}

[Expand]\scriptstyle\frac{1}{9}\times\frac{1}{7}

[Expand]\scriptstyle\frac{4}{7}\times\frac{5}{6}

[Expand]\scriptstyle\frac{4}{7}\times\frac{5}{9}

[Expand]\scriptstyle\frac{4}{7}\times\frac{7}{9}

[Expand]\scriptstyle\frac{7}{9}\times\frac{4}{7}

[Expand]\scriptstyle\frac{7}{8}\times\frac{9}{10}

[Expand]\scriptstyle\frac{2}{11}\times\frac{2}{13}

[Expand]\scriptstyle\frac{10}{11}\times\frac{12}{13}

[Expand]\scriptstyle\frac{4}{5}\times\frac{3}{13}

[Expand]\scriptstyle\frac{9}{15}\times\frac{11}{17}

[Expand]\scriptstyle\frac{9}{13}\times\frac{17}{19}

[Expand]\scriptstyle\frac{20}{3}\times\frac{19}{3}

multiplication of fraction of fraction by fraction of fraction

[Expand]\scriptstyle\left(\frac{2}{5}\sdot\frac{1}{10}\right)\times\left(\frac{2}{8}\sdot\frac{1}{3}\right)

[Expand]\scriptstyle\left(\frac{3}{4}\sdot\frac{3}{5}\right)\times\left(\frac{5}{6}\sdot\frac{3}{7}\right)

multiplication of fraction of integer and fraction by fraction of integer and fraction

[Expand]\scriptstyle\left[\frac{7}{8}\sdot\left(5+\frac{1}{3}\right)\right]\times\left[\frac{3}{5}\sdot\left(6+\frac{1}{4}\right)\right]

[Expand]\scriptstyle\left[\frac{2}{3}\sdot\left(5+\frac{5}{6}\right)\right]\times\left[\frac{5}{7}\sdot\left(8+\frac{4}{9}\right)\right]

multiplication of fraction by integer or integer by fraction

[Expand]\scriptstyle\frac{1}{3}\times1

[Expand]\scriptstyle\frac{1}{3}\times2

[Expand]\scriptstyle2\times\frac{3}{4}

[Expand]\scriptstyle4\times\frac{2}{3}

[Expand]\scriptstyle4\times\frac{3}{5}

[Expand]\scriptstyle5\times\frac{4}{6}

[Expand]\scriptstyle\frac{3}{4}\times9

[Expand]\scriptstyle\frac{5}{6}\times10

[Expand]\scriptstyle\frac{5}{7}\times40

multiplication of fraction of fraction of integer by fraction of fraction of integer

[Expand]\scriptstyle\left[\left(\frac{3}{4}\sdot\frac{1}{5}\right)\sdot8\right]\times\left[\left(\frac{5}{6}\sdot\frac{1}{7}\right)\sdot15\right]

[Expand]\scriptstyle\left[\left(\frac{2}{3}\sdot\frac{3}{4}\right)\sdot6\right]\times\left[\left(\frac{3}{4}\sdot\frac{4}{7}\right)\sdot8\right]

[Expand]\scriptstyle\left(\frac{2}{4}\times\frac{2}{6}\right)+\left(\frac{4}{5}\times\frac{5}{6}\right)


multiplication of sexagesimal fractions

[Expand]\scriptstyle\left(2+24^\prime+43^{\prime\prime}\right)\times\left(3+3^\prime+8^{\prime\prime}\right)

addition of fractions

[Expand]\scriptstyle\frac{1}{2}+\frac{1}{3}

[Expand]\scriptstyle\frac{1}{3}+\frac{1}{2}

[Expand]\scriptstyle\frac{1}{3}+\frac{1}{4}

[Expand]\scriptstyle\frac{1}{3}+\frac{1}{5}

[Expand]\scriptstyle\frac{1}{4}+\frac{1}{5}

[Expand]\scriptstyle\frac{1}{3}+\frac{1}{7}

[Expand]\scriptstyle\frac{1}{2}+\frac{1}{3}+\frac{1}{4}

[Expand]\scriptstyle\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}

[Expand]\scriptstyle\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}

[Expand]\scriptstyle\frac{2}{3}+\frac{3}{4}

[Expand]\scriptstyle\frac{3}{4}+\frac{4}{5}

[Expand]\scriptstyle\frac{3}{4}+\frac{5}{6}

[Expand]\scriptstyle\frac{3}{5}+\frac{4}{7}

[Expand]\scriptstyle\frac{3}{7}+\frac{2}{3}

[Expand]\scriptstyle\frac{3}{8}+\frac{7}{10}

[Expand]\scriptstyle\frac{2}{3}+\frac{3}{4}+\frac{4}{5}

[Expand]\scriptstyle\frac{3}{4}+\frac{4}{5}+\frac{5}{6}

[Expand]\scriptstyle\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}

[Expand]\scriptstyle\left(\frac{3}{4}+\frac{4}{5}\right)+\left(\frac{5}{6}+\frac{6}{7}\right)

[Expand]\scriptstyle\frac{2}{3}+\frac{4}{5}+\frac{5}{6}+\left(\frac{3}{4}\sdot\frac{1}{6}\right)

[Expand]\scriptstyle\left[\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]+\left(\frac{7}{8}+\frac{8}{9}\right)

[Expand]\scriptstyle\left[\frac{5}{11}+\left(\frac{3}{4}\sdot\frac{1}{11}\right)\right]+\left[\frac{10}{13}+\left(\frac{8}{9}\sdot\frac{1}{13}\right)\right]

[Expand]\scriptstyle\left(3+\frac{1}{2}\right)+\frac{2}{5}

[Expand]\scriptstyle\left(2+\frac{1}{2}\right)+\left(3+\frac{1}{4}\right)

[Expand]\scriptstyle\left(2+\frac{1}{2}\right)+\left(3+\frac{2}{5}\right)

[Expand]\scriptstyle\left[8+\frac{5}{6}+\left(\frac{1}{2}\sdot\frac{1}{6}\right)\right]+\left(6+\frac{5}{7}+\frac{10}{11}\right)

[Expand]\scriptstyle\frac{5}{9}+\left(\frac{4}{5}\sdot\frac{1}{7}\right)

[Expand]\scriptstyle\left(\frac{3}{4}\sdot5\right)+6

[Expand]\scriptstyle\left(\frac{2}{3}\sdot7\right)+\left(\frac{7}{8}\sdot9\right)

[Expand]\scriptstyle\left[\frac{3}{4}\sdot\left(5+\frac{1}{2}\right)\right]+\left[\frac{5}{6}\sdot\left(6+\frac{1}{3}\right)\right]

[Expand]\scriptstyle\left[2+\frac{3}{8}+\left(\frac{2}{4}\sdot\frac{1}{8}\right)\right]+\frac{4}{5}+\left[\frac{6}{7}+\left(\frac{3}{8}\sdot\frac{1}{7}\right)\right]

[Expand]\scriptstyle\left(\frac{2}{5}\sdot\frac{2}{9}\sdot2\right)+\left[\left[\frac{1}{8}+\left(\frac{2}{9}\sdot\frac{1}{7}\sdot\frac{1}{8}\right)\right]\sdot\left[\frac{1}{4}+\left(\frac{2}{6}\sdot\frac{1}{4}\right)\right]\right]

subtraction of fractions

[Expand]\scriptstyle\frac{1}{3}-\frac{1}{4}

[Expand]\scriptstyle\frac{1}{4}-\frac{1}{5}

[Expand]\scriptstyle\frac{2}{4}-\frac{1}{5}

[Expand]\scriptstyle\frac{3}{4}-\frac{1}{2}

[Expand]\scriptstyle\frac{3}{4}-\frac{2}{3}

[Expand]\scriptstyle\frac{1}{3}-\frac{1}{5}

[Expand]\scriptstyle\frac{1}{3}-\frac{1}{7}

[Expand]\scriptstyle\frac{2}{3}-\frac{3}{7}

[Expand]\scriptstyle\frac{3}{4}-\frac{2}{8}

[Expand]\scriptstyle\frac{2}{5}-\frac{3}{8}

[Expand]\scriptstyle\frac{5}{7}-\frac{4}{9}

[Expand]\scriptstyle\left(\frac{1}{3}+\frac{1}{4}\right)-\left(\frac{1}{5}+\frac{1}{6}\right)

[Expand]\scriptstyle\left(2+\frac{1}{2}\right)-\frac{3}{4}

[Expand]\scriptstyle\left(3+\frac{3}{4}\right)-\left(2+\frac{1}{2}\right)

[Expand]\scriptstyle\left[\left[\frac{8}{9}+\left(\frac{3}{7}\sdot\frac{1}{5}\sdot\frac{1}{9}\right)\right]\sdot\left(\frac{5}{6}\sdot3\right)\right]-\left[\left[\frac{3}{4}+\left(\frac{2}{5}\sdot\frac{1}{4}\right)\right]\sdot\frac{2}{9}\right]

[Expand]\scriptstyle\left[3+\frac{5}{9}+\left(\frac{3}{7}\sdot\frac{1}{9}\right)+\left(\frac{2}{4}\sdot\frac{1}{7}\sdot\frac{1}{9}\right)\right]-\left[\frac{7}{9}+\left(\frac{5}{7}\sdot\frac{1}{9}\right)+\left(\frac{3}{4}\sdot\frac{1}{7}\sdot\frac{1}{9}\right)\right]

Rule of Three

[Expand]\scriptstyle2:4=4:8

[Expand]\scriptstyle2:3=5:X

[Expand]\scriptstyle5:7=10:X

[Expand]\scriptstyle3:7=5:X

[Expand]\scriptstyle3:7=X:\left(11+\frac{2}{3}\right)

[Expand]\scriptstyle5:\left(7+\frac{1}{2}\right)=2:X

[Expand]\scriptstyle\left(5+\frac{2}{7}\right):4=20:X

[Expand]\scriptstyle9:\frac{2}{3}=8:X

[Expand]\scriptstyle\frac{2}{3}:8=9:X

[Expand]\scriptstyle\frac{1}{3}:\frac{1}{4}=\frac{1}{5}:X

[Expand]\scriptstyle\frac{3}{7}:\frac{8}{9}=\frac{4}{5}:X

[Expand]\scriptstyle\frac{2}{3}:\frac{4}{9}=\frac{4}{13}:X

[Expand]\scriptstyle\frac{2}{3}:\left(4+\frac{1}{2}\right)=6:X

[Expand]\scriptstyle6:\left(40+\frac{1}{2}\right)=\frac{2}{3}:X

[Expand]\scriptstyle20:\left(15+\frac{5}{37}\right)=\left(5+\frac{2}{7}\right):X

[Expand]\scriptstyle\frac{2}{3}:\left(7+\frac{4}{9}\right)=\frac{4}{13}:X

[Expand]\scriptstyle\left(3+\frac{1}{3}\right):\left(4+\frac{1}{4}\right)=\left(5+\frac{1}{5}\right):X

[Expand]\scriptstyle\left(5+\frac{2}{3}\right):\left(6+\frac{3}{4}\right)=\left(8+\frac{5}{12}\right):X

[Expand]\scriptstyle\left[\frac{3}{4}\sdot\left(3-\frac{1}{4}\right)\right]:\left[\frac{4}{5}\sdot\left(5-\frac{1}{5}\right)\right]=\left[\frac{5}{6}\sdot\left(6-\frac{1}{6}\right)\right]:X

Roots

Extraction of Roots

[Expand]\scriptstyle\sqrt{\frac{2}{8}}

[Expand]\scriptstyle\sqrt{100}

[Expand]\scriptstyle\sqrt{400}

[Expand]\scriptstyle\sqrt{225}

[Expand]\scriptstyle\sqrt{5625}

[Expand]\scriptstyle\sqrt{7056}

[Expand]\scriptstyle\sqrt{10375}

[Expand]\scriptstyle\sqrt{164960}

[Expand]\scriptstyle\sqrt{456789}

[Expand]\scriptstyle\sqrt{583696}

[Expand]\scriptstyle\sqrt{824464}

[Expand]\scriptstyle\sqrt{5499025}

[Expand]\scriptstyle\sqrt{6169002849}

[Expand]\scriptstyle\sqrt{344680129066}

Extraction of Cube Roots

[Expand]\scriptstyle\sqrt[3]{\frac{1}{2}}

[Expand]\scriptstyle\sqrt[3]{1+\frac{1}{2}}

[Expand]\scriptstyle\sqrt[3]{10}

[Expand]\scriptstyle\sqrt[3]{15}

[Expand]\scriptstyle\sqrt[3]{12167}

[Expand]\scriptstyle\sqrt[3]{571787}

[Expand]\scriptstyle\sqrt[3]{12812904}

Approximation

[Expand]\scriptstyle\sqrt{2}

[Expand]\scriptstyle\sqrt{20}

[Expand]\scriptstyle\sqrt{200}

[Expand]\scriptstyle\sqrt{2000}

[Expand]\scriptstyle\sqrt{20000}

[Expand]\scriptstyle\sqrt{4000}

[Expand]\scriptstyle\sqrt{5}

[Expand]\scriptstyle\sqrt{10}

[Expand]\scriptstyle\sqrt{15}

[Expand]\scriptstyle\sqrt{6}

[Expand]\scriptstyle\sqrt{7}

[Expand]\scriptstyle\sqrt{18}

[Expand]\scriptstyle\sqrt{29}

[Expand]\scriptstyle\sqrt{925}

[Expand]\scriptstyle\sqrt{76543}

[Expand]\scriptstyle\sqrt{6169004404}

[Expand]\scriptstyle\sqrt{\frac{4}{6}\sdot\left(4+\frac{5}{9}\right)}

Addition of Roots

[Expand]\scriptstyle\sqrt{9}+\sqrt{4}


[Expand]\scriptstyle\sqrt{10}+\sqrt{2}


[Expand]\scriptstyle\sqrt{3}+\sqrt{12}


[Expand]\scriptstyle\sqrt{6}+\sqrt{7}


[Expand]\scriptstyle\sqrt{12}+\sqrt{48}


[Expand]\scriptstyle\sqrt{18}+\sqrt{8}


[Expand]\scriptstyle\sqrt{8}+\sqrt{19}


[Expand]\scriptstyle\sqrt[3]{96}+\sqrt[3]{324}


[Expand]\scriptstyle\left(4+\sqrt{12}\right)+\left(5+\sqrt{3}\right)


[Expand]\scriptstyle\left(4+\sqrt{3}\right)+\left(\sqrt{12}-3\right)


[Expand]\scriptstyle\left(4+\sqrt{3}\right)+\left(\sqrt{12}-2\right)


[Expand]\scriptstyle\left(4-\sqrt{3}\right)+\left(\sqrt{12}-2\right)


[Expand]\scriptstyle\sqrt{3}+\sqrt{6}+\sqrt{12}+\sqrt{24}


Subtraction of Roots

[Expand]\scriptstyle\sqrt{9}-\sqrt{4}


[Expand]\scriptstyle\sqrt{16}-\sqrt{9}


[Expand]\scriptstyle\sqrt{12}-\sqrt{3}


[Expand]\scriptstyle\sqrt{18}-\sqrt{8}


[Expand]\scriptstyle\sqrt{7}-\sqrt{6}


[Expand]\scriptstyle19-\left(10-\sqrt{12}\right)


[Expand]\scriptstyle10-\left(24-\sqrt{250}\right)


[Expand]\scriptstyle16-\left(8+\sqrt{50}\right)


[Expand]\scriptstyle\left(13-\sqrt{20}\right)-\left(6-\sqrt{5}\right)


Multiplication of Roots

[Expand]\scriptstyle\sqrt{9}\times\sqrt{4}


[Expand]\scriptstyle\sqrt{4}\times\sqrt{9}


[Expand]\scriptstyle\sqrt{7}\times\sqrt{8}


[Expand]\scriptstyle\sqrt{5}\times\sqrt{12}


[Expand]\scriptstyle2\times\sqrt{16}


[Expand]\scriptstyle\frac{1}{2}\times\sqrt{9}


[Expand]\scriptstyle\frac{2}{3}\times\sqrt{9}


[Expand]\scriptstyle\sqrt{6}\times3


[Expand]\scriptstyle\sqrt{7}\times3


[Expand]\scriptstyle\sqrt{7}\times8


[Expand]\scriptstyle\sqrt{5}\times\left(\sqrt{7}+4\right)


[Expand]\scriptstyle\sqrt{3}\times\left(6-\sqrt{8}\right)


[Expand]\scriptstyle\left(3+\sqrt{5}\right)\times\left(3+\sqrt{5}\right)


[Expand]\scriptstyle\left(5+\sqrt{6}\right)\times\left(5+\sqrt{6}\right)


[Expand]\scriptstyle\left(3+\sqrt{5}\right)\times\left(4+\sqrt{7}\right)


[Expand]\scriptstyle\left(3+\sqrt{4}\right)\times\left(4+\sqrt{9}\right)


[Expand]\scriptstyle\left(3-\sqrt{5}\right)\times\left(4-\sqrt{7}\right)


[Expand]\scriptstyle\left(3-\sqrt{5}\right)\times\left(3-\sqrt{5}\right)


[Expand]\scriptstyle\left(5+\sqrt{3}\right)\times\left(5-\sqrt{3}\right)


[Expand]\scriptstyle\left(3+\sqrt{4}\right)\times\left(5-\sqrt{9}\right)


[Expand]\scriptstyle\sqrt{8}\times\left(\sqrt{8}-2\right)


[Expand]\scriptstyle\left(\sqrt{8}-2\right)\times\left(\sqrt{10}-3\right)


[Expand]\scriptstyle\left(\sqrt{12}-2\right)\times\left(\sqrt{12}-2\right)


[Expand]\scriptstyle\left(\sqrt{32}-3\right)\times\left(\sqrt{32}-3\right)


[Expand]\scriptstyle\left(\sqrt{15}-3\right)\times\left(\sqrt{12}+2\right)


[Expand]\scriptstyle\left(\sqrt{8}+2\right)\times\left(\sqrt{8}-2\right)


[Expand]\scriptstyle\left(2\times\sqrt{10}\right)\times\left(\frac{1}{2}\times\sqrt{5}\right)


[Expand]\scriptstyle\sqrt{5}\times\left(\sqrt{7}+\sqrt{10}\right)


[Expand]\scriptstyle\sqrt{5}\times\left(\sqrt{12}-\sqrt{8}\right)


[Expand]\scriptstyle\left(\sqrt{5}+\sqrt{7}\right)\times\left(\sqrt{10}+\sqrt{15}\right)


[Expand]\scriptstyle\left(\sqrt{5}+\sqrt{7}\right)\times\left(\sqrt{5}+\sqrt{7}\right)


[Expand]\scriptstyle\left(\sqrt{5}+\sqrt{7}\right)\times\left(\sqrt{10}-\sqrt{6}\right)


[Expand]\scriptstyle\left(\sqrt{10}+\sqrt{7}\right)\times\left(\sqrt{10}-\sqrt{7}\right)


[Expand]\scriptstyle\left(\sqrt{8}+\sqrt{4}\right)\times\left(\sqrt{8}-\sqrt{4}\right)


[Expand]\scriptstyle\left(\sqrt{48}+\sqrt{10}\right)\times\left(\sqrt{48}-\sqrt{10}\right)


[Expand]\scriptstyle\left(\sqrt{12}-\sqrt{7}\right)\times\left(\sqrt{15}-\sqrt{10}\right)


[Expand]\scriptstyle\left(\sqrt{12}-\sqrt{7}\right)\times\left(\sqrt{12}-\sqrt{7}\right)


[Expand]\scriptstyle3\times\sqrt{4}


[Expand]\scriptstyle3\times\sqrt[3]{8}


[Expand]\scriptstyle3\times\sqrt[3]{5}


[Expand]\scriptstyle\sqrt{4}\times\sqrt[3]{8}


[Expand]\scriptstyle\sqrt{9}\times\sqrt[3]{8}


[Expand]\scriptstyle\sqrt[3]{5}\times\sqrt[3]{6}


[Expand]\scriptstyle2\times\sqrt[4]{5}


[Expand]\scriptstyle\sqrt[3]{8}\times\sqrt[4]{16}


[Expand]\scriptstyle\sqrt[3]{3}\times\sqrt[4]{4}


[Expand]\scriptstyle\sqrt[4]{4}\times\sqrt[4]{7}


Division of Roots

[Expand]\scriptstyle\sqrt{9}\div\sqrt{4}


[Expand]\scriptstyle\sqrt{4}\div\sqrt{9}


[Expand]\scriptstyle\sqrt{10}\div\sqrt{2}


[Expand]\scriptstyle\sqrt{30}\div\sqrt{6}


[Expand]\scriptstyle4\div\sqrt{9}


[Expand]\scriptstyle12\div\sqrt{4}


[Expand]\scriptstyle20\div\sqrt{10}


[Expand]\scriptstyle8\div\left(3+\sqrt{4}\right)


[Expand]\scriptstyle19\div\left(2+\sqrt{16}\right)


[Expand]\scriptstyle\sqrt{64}\div\left(\sqrt{8}-\sqrt{4}\right)


[Expand]\scriptstyle\sqrt{64}\div\left(\sqrt{8}+\sqrt{4}\right)


[Expand]\scriptstyle\left(5+\sqrt{16}\right)\div3


[Expand]\scriptstyle20\div\left(4-\sqrt{9}\right)


[Expand]\scriptstyle\left(19+\sqrt{25}\right)\div\left(5+\sqrt{9}\right)


[Expand]\scriptstyle\left(2\times\sqrt{20}\right)\div\left(3\times\sqrt{6}\right)


[Expand]\scriptstyle36\div\left(\sqrt{4}+\sqrt{9}+\sqrt{16}\right)


[Expand]\scriptstyle70\div\left(\sqrt{4}+\sqrt{9}+\sqrt{16}+\sqrt{25}\right)


[Expand]\scriptstyle\sqrt{6}\div\sqrt[3]{10}


[Expand]\scriptstyle\sqrt[3]{18}\div\sqrt[4]{10}


Multiplication of Algebraic Species

[Expand]\scriptstyle3x\times6


[Expand]\scriptstyle2x\times2x


[Expand]\scriptstyle\left(10+x\right)\times x


[Expand]\scriptstyle\left(10-x\right)\times x


[Expand]\scriptstyle\left(10+x\right)\times\left(10+x\right)


[Expand]\scriptstyle\left(10-x\right)\times\left(10-x\right)


[Expand]\scriptstyle\left(10+x\right)\times\left(10-x\right)


[Expand]\scriptstyle\left(10+x\right)\times\left(x-10\right)


[Expand]\scriptstyle\left(10+\frac{2}{3}x\right)\times\left(3-6x\right)


Linear Equation

[Expand]\scriptstyle bx=\sqrt[3]{c}


[Expand]\scriptstyle c=\sqrt[3]{bx}



Quadratic Equation

ax²=bx

[Expand]squares equal roots \scriptstyle ax^2=bx


[Expand]\scriptstyle x^2=5x


[Expand]\scriptstyle\frac{1}{2}x^2=10x


[Expand]\scriptstyle5x^2=20x


ax²=c

[Expand]squares equal numbers \scriptstyle ax^2=c


[Expand]\scriptstyle x^2=16


[Expand]\scriptstyle5x^2=45


[Expand]\scriptstyle\frac{1}{3}x^2=27


bx=c

[Expand]roots equal numbers \scriptstyle bx=c

ax²+bx=c

[Expand]\scriptstyle ax^2+bx=c


[Expand]\scriptstyle x^2+10x=39


[Expand]\scriptstyle2x^2+10x=48


[Expand]\scriptstyle3x^2+15x=72


[Expand]\scriptstyle\frac{1}{2}x^2+5x=28


ax²+c=bx

[Expand]\scriptstyle ax^2+c=bx


[Expand]\scriptstyle x^2+21=10x


[Expand]\scriptstyle x^2+25=10x


bx+c=ax²

[Expand]\scriptstyle bx+c=ax^2


[Expand]\scriptstyle3x+4=x^2


Compound Quadratic Equations

[Expand]\scriptstyle ax^2=\sqrt[3]{c}


[Expand]\scriptstyle c=\sqrt[3]{ax^2}


[Expand]\scriptstyle\left[x^2-\left(\frac{1}{3}x^2+2\right)\right]^2=x^2+24


[Expand]\scriptstyle3\sqrt{x^2}+4\sqrt{x^2-3\sqrt{x^2}}=20


[Expand]\scriptstyle\left(x^2-\frac{1}{3}x^2\right)\sdot3\sqrt{x^2}=x^2


[Expand]\scriptstyle\left(x^2-\frac{1}{3}x^2\right)\sdot3\sqrt{x^2-\frac{1}{3}x^2}=x^2


[Expand]\scriptstyle3\sqrt{x^2}+2\sqrt{x^2-3\sqrt{x^2}}=x^2


[Expand]\scriptstyle3\sqrt{x^2}+4\sqrt{x^2-3\sqrt{x^2}}=x^2+4


[Expand]\scriptstyle x^2\sdot\left(x^2+\sqrt{10}\right)=9x^2


[Expand]\scriptstyle\left(\sqrt{8x^2}\sdot\sqrt{3x^2}\right)+20=\left(x^2\right)^2


[Expand]\scriptstyle\left(\sqrt{6x^2}\sdot\sqrt{5x^2}\right)+10x^2+20=\left(x^2\right)^2


[Expand]\scriptstyle\left(x^2+10\right)\sdot\sqrt{5}=\left(x^2\right)^2


[Expand]\scriptstyle\left(\sqrt{x^2\sdot2x^2}+2\right)\sdot x^2=30



[Expand]\scriptstyle\left[x^2-\left(2\sqrt{x^2}+10\right)\right]^2=8x^2


[Expand]\scriptstyle2\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}+\sqrt{\frac{1}{3}x^2}=x^2


[Expand]\scriptstyle2\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}+\sqrt{\frac{1}{3}x^2}=20


[Expand]\scriptstyle x^2+4\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}+\sqrt{\frac{1}{3}x^2}=10


[Expand]\scriptstyle\left(x^2+\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}\right)^2=5x^2


[Expand]\scriptstyle\left(x^2+\sqrt{x^2}+\sqrt{\frac{1}{2}x^2}\right)^2=20


[Expand]\scriptstyle\left(x^2+\sqrt{\frac{1}{2}x^2}\right)^2=4x^2


[Expand]\scriptstyle\left(x^2+7\right)\sdot\sqrt{3x^2}=10x^2


[Expand]\scriptstyle\left(x^2+\sqrt{3x^2}\right)\sdot\sqrt{{\color{red}{2}}x^2}=4x^2


[Expand]\scriptstyle\left(\sqrt{\frac{1}{2}x^2}+3\right)\sdot\left(\sqrt{\frac{1}{3}x^2}+2\right)=20


[Expand]\scriptstyle\frac{\sqrt{10}x^2}{2+\sqrt{3}}=x^2-10


[Expand]\scriptstyle\sqrt{x^2}+\sqrt{\sqrt{x^2}}+\sqrt{2\sqrt{x^2}}+\sqrt{5x^2}=10


quadratic equation in two variables

[Expand]\scriptstyle\begin{cases}\scriptstyle b^2=3a^2\\\scriptstyle\left(a^2+\sqrt{a^2}\right)\sdot\left(b^2+\sqrt{b^2}\right)=10b^2\end{cases}


quadratic equation in three variables

[Expand]\scriptstyle\begin{cases}\scriptstyle a^2+b^2=c^2\\\scriptstyle a\sdot c=b^2\\\scriptstyle a\sdot b=10\end{cases}


Cubic Equation

[Expand]\scriptstyle ax^3=\sqrt[3]{c}


Biquadratic Equation

[Expand]\scriptstyle4\sdot\left(x^2+8\right)=x^4


[Expand]\scriptstyle c=ax^4+\sqrt{bx^4}


[Expand]\scriptstyle ax^4+bx^2=c


[Expand]\scriptstyle bx^2=ax^4+c


[Expand]\scriptstyle ax^4=bx^2+c


indeterminate equation

[Expand]\scriptstyle x^2+5=n^2

[Expand]\scriptstyle x^2-10=n^2

[Expand]\scriptstyle x^2+3x=n^2

[Expand]\scriptstyle x^2-6x=n^2

[Expand]\scriptstyle x^2+10x+20=n^2

[Expand]\scriptstyle x^2-8x-30=n^2


Euclid

Elements - Introduction

[Expand]common notions

Elements II

Elements II-1

[Expand]proposition

[Expand]\scriptstyle a\sdot\left(\sum_{i=1}^n b_i\right)=\sum_{i=1}^n \left(a\sdot b_i\right)

Examples

[Expand]\scriptstyle\left(2\sdot4\right)+\left(3\sdot4\right)+\left(5\sdot4\right)=10\sdot4

[Expand]\scriptstyle\left(2\sdot10\right)+\left(3\sdot10\right)+\left(5\sdot10\right)+\left(2\sdot10\right)=12\sdot10

Elements II-2

[Expand]\scriptstyle\sum_{k=1}^n \left[\left(\sum_{i=1}^n a_i\right)\sdot a_k\right]=\left(\sum_{i=1}^n a_i\right)^2

Examples

[Expand]\scriptstyle\left(7\sdot10\right)+\left(3\sdot10\right)=10^2

[Expand]\scriptstyle\left(3\sdot12\right)+\left(4\sdot12\right)+\left(5\sdot12\right)=12^2

Elements II-3

[Expand]\scriptstyle\left(a+b\right)\sdot b=\left(a\sdot b\right)+b^2

Examples

[Expand]\scriptstyle10\sdot3=\left(3\sdot7\right)+3^2

[Expand]\scriptstyle12\sdot8=\left(4\sdot8\right)+8^2

Elements II-4

[Expand]\scriptstyle\left(a+b\right)^2=a^2+b^2+2\sdot\left(a\sdot b\right)

Examples

[Expand]\scriptstyle10^2=3^2+7^2+2\sdot\left(3\sdot7\right)

[Expand]\scriptstyle12^2=4^2+8^2+2\sdot\left(4\sdot8\right)

Elements II-5

[Expand]\scriptstyle\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2=\left(b\sdot b\right)+\left[b-\left[\frac{1}{2}\sdot\left(a\sdot b\right)\right]\right]^2=\left(b\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a\sdot b\right)\right]-a\right]^2

Examples

[Expand]\scriptstyle5^2=\left(3\sdot7\right)+\left(7-5\right)^2

[Expand]\scriptstyle6^2=\left(4\sdot8\right)+\left(8-6\right)^2

Elements II-6

[Expand]\scriptstyle\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2

Examples

[Expand]\scriptstyle\left[\left(10+2\right)\sdot2\right]+\left(\frac{1}{2}\sdot10\right)^2=\left[\left(\frac{1}{2}\sdot10\right)+2\right]^2

[Expand]\scriptstyle\left[\left(12+4\right)\sdot4\right]+\left(\frac{1}{2}\sdot12\right)^2=\left[\left(\frac{1}{2}\sdot12\right)+4\right]^2

Elements II-7

[Expand]\scriptstyle\left(a+b\right)^2+a^2=\left[2\sdot\left(a+b\right)\sdot a\right]+b^2

Examples

[Expand]\scriptstyle10^2+7^2=\left(2\sdot10\sdot7\right)+3^2

[Expand]\scriptstyle12^2+8^2=\left(2\sdot12\sdot8\right)+4^2

Elements II-8

[Expand]\scriptstyle4\sdot\left[\left(a+b\right)\sdot a\right]+b^2=\left[\left(a+b\right)+a\right]^2

Examples

[Expand]\scriptstyle4\sdot\left(10\sdot7\right)+3^2=\left(10+7\right)^2

[Expand]\scriptstyle4\sdot\left(12\sdot4\right)+8^2=\left(12+4\right)^2

Elements II-9

[Expand]\scriptstyle a^2+b^2=2\sdot\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2+\left[a-\left[\frac{1}{2}\sdot\left(a+b\right)\right]\right]^2\right]

Examples

[Expand]\scriptstyle 7^2+3^2=2\sdot\left[\left(\frac{1}{2}\sdot10\right)^2+\left[7-\left(\frac{1}{2}\sdot10\right)\right]^2\right]

[Expand]\scriptstyle 8^2+4^2=2\sdot\left[\left(\frac{1}{2}\sdot12\right)^2+\left[8-\left(\frac{1}{2}\sdot12\right)\right]^2\right]

Elements II-10

[Expand]\scriptstyle\left(a+b\right)^2+b^2=2\sdot\left[\left(\frac{1}{2}\sdot a\right)^2+\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2\right]

Examples

[Expand]\scriptstyle\left(10+2\right)^2+2^2=2\sdot\left[\left(\frac{1}{2}\sdot10\right)^2+\left[\left(\frac{1}{2}\sdot10\right)+2\right]^2\right]

[Expand]\scriptstyle\left(12+8\right)^2+8^2=2\sdot\left[\left(\frac{1}{2}\sdot12\right)^2+\left[\left(\frac{1}{2}\sdot12\right)+8\right]^2\right]

Elements V

[Expand]Elements V-9


Elements V-15

[Expand]\scriptstyle\left(n\sdot a\right):\left(n\sdot b\right)=a:b

Elements VIII

[Expand]Elements VIII-11

Elements IX

[Expand]Elements IX-21

[Expand]Elements IX-22

[Expand]Elements IX-23

[Expand]Elements IX-28

[Expand]Elements IX-29

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