Student Basics - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Student "Basics" Package

 

The new Student Basics package helps to explore the foundations of higher math, making it possible to provide step-by-step breakdowns for expanding and simplifying mathematical expressions, such as simplifying fractions, expanding products of polynomials, or solving linear equations. All the steps to the solution are shown and documented, so that a student can easily understand what is happening at each stage of the solution. Students can use this package to understand where results are coming from and learn how to solve these problems on their own.

 

Here are 101 interesting examples showing the steps involved to solve or expand:

 

withStudent:-Basics:

 

LinearSolveStepsx + 3 = 8, x;

x+3=8x=8−3subtract from both sidesx=5add terms

(1)

LinearSolveSteps5⋅x−2 = 13, x;

5⁢x−2=135⁢x=13+2subtract from both sidesx=13+25divide both sidesx=155add termsx=3reduce fraction by gcd

(2)

LinearSolveSteps4x + 5 = x - 4, x, implicitmultiply;

4⁢x+5=x−44⁢x−x=−4−5subtract from both sides3⁢x=−4−5add terms3⁢x=−9add termsx=−93divide both sidesx=−3reduce fraction by gcd

(3)

LinearSolveSteps3(n - 1.8) = 2n + 1, n, implicitmultiply;

3⁢n−1.8=2⁢n+13⁢n+3⁢−1.8=2⁢n+1distributive multiply3⁢n−5.4=2⁢n+1multiply constants3⁢n−2⁢n=1+5.4subtract from both sidesn=1+5.4add termsn=6.4add terms

(4)

LinearSolveSteps7y + 5 - 3y + 1 = 2y + 2 , y, implicitmultiply;

7⁢y+5−3⁢y+1=2⁢y+27⁢y−3⁢y−2⁢y=2−5−1subtract from both sides2⁢y=2−5−1add terms2⁢y=−4add termsy=−42divide both sidesy=−2reduce fraction by gcd

(5)

LinearSolveSteps2x + 4 = 10, x, implicitmultiply;

2⁢x+4=102⁢x=10−4subtract from both sidesx=10−42divide both sidesx=62add termsx=3reduce fraction by gcd

(6)

LinearSolveSteps3x - 4 = -10, x, implicitmultiply; 

3⁢x−4=−103⁢x=−10+4subtract from both sidesx=−10+43divide both sidesx=−63add termsx=−2reduce fraction by gcd

(7)

LinearSolveSteps4x - 4y = 8, x, implicitmultiply; 

4⁢x−4⁢y=84⁢x=8+4⁢ysubtract from both sidesx=8+4⁢y4divide both sidesx=4⁢2+y4factorx=2+ydivide

(8)

LinearSolveStepsx + 3^2 = 12 , x;

x+32=12x=12−32subtract from both sidesx=12−9evaluate powerx=3add terms

(9)

LinearSolveStepsx + y2 = 12, x;

y2+x=12x=12−y2subtract from both sidesx=−y2+12reorder terms

(10)

LinearSolveSteps x2 + y24 = 14⋅ x2 − 2⋅x + 14, x 

x24+y24=x24+−2⁢x+14x24−x24−−2⁢x=14−y24subtract from both sidesx24−x24+2⁢x=14−y24distribute negation2⁢x=14−y24add termsx=14−y242divide both sides

(11)

LinearSolveSteps4(8-3x) = 32 - 8(x+2), x, implicitmultiply;

4⁢8−3⁢x=32−8⁢x+24·8+4⁢−3⁢x=32−8⁢x+2distributive multiply32+4⁢−3⁢x=32−8⁢x+2multiply constants32+−12⁢x=32−8⁢x+2multiply constants−12⁢x+8⁢x+2=32−32subtract from both sides−12⁢x+8⁢x+8·2=32−32distributive multiply−12⁢x+8⁢x+16=32−32multiply constants−4⁢x+16=32−32add terms−4⁢x+16=0add terms−4⁢x=−16subtract from both sidesx=−16−4divide both sidesx=4reduce fraction by gcd

(12)

LinearSolveSteps2x/3 = -18, x, implicitmultiply;

2⁢x3=−182⁢x=3⁢−18multiply rhs by denominator of lhsx=3⁢−182divide both sidesx=−542multiply constantsx=−27reduce fraction by gcd

(13)

LinearSolveSteps2/(3x) = -18, x, implicitmultiply;

23⁢x=−1812⁢3⁢x=−118reciprocal of both sidesx=−11832divide both sidesx=−118⁢23rewrite division as multiplication by reciprocalx=−127multiply fraction and reduce by gcd

(14)

LinearSolveSteps(2x)/(3x^2) = -18, x, implicitmultiply;

2⁢x3⁢x2=−1823⁢x=−18divide out common terms3⁢x2=−118reciprocal of both sidesx=−11832divide both sidesx=−118⁢23rewrite division as multiplication by reciprocalx=−127multiply fraction and reduce by gcd

(15)

LinearSolveSteps2/(3x) = -18, x, implicitmultiply;

23⁢x=−1812⁢3⁢x=−118reciprocal of both sidesx=−11832divide both sidesx=−118⁢23rewrite division as multiplication by reciprocalx=−127multiply fraction and reduce by gcd

(16)

LinearSolveSteps(2x)/(3x^2) = -18, x, implicitmultiply; 

2⁢x3⁢x2=−1823⁢x=−18divide out common terms3⁢x2=−118reciprocal of both sidesx=−11832divide both sidesx=−118⁢23rewrite division as multiplication by reciprocalx=−127multiply fraction and reduce by gcd

(17)

LinearSolveStepsx/3-5/12 = 3/4+1/2x, x, implicitmultiply;

x3−512=34+12⁢xx3−12⁢x=34+512subtract from both sidesx3−x2=34+512multiply fraction−x6=34+512add terms−x6=76add terms−x=6⁢76multiply rhs by denominator of lhs−x=426multiply fraction−x=7reduce fraction by gcdx=−7negate both sides

(18)

LinearSolveSteps10-3x = 7, x, implicitmultiply; 

10−3⁢x=7−3⁢x=7−10subtract from both sides−3⁢x=−3add termsx=−3−3divide both sidesx=1divide out common terms

(19)

LinearSolveSteps2(x+5)-7 = 3(x-2), x, implicitmultiply;

2⁢x+5−7=3⁢x−22⁢x+5−7=3⁢x+3⁢−2distributive multiply2⁢x+5−7=3⁢x−6multiply constants2⁢x+5−3⁢x=−6+7subtract from both sides2⁢x+2·5−3⁢x=−6+7distributive multiply2⁢x+10−3⁢x=−6+7multiply constants−x+10=−6+7add terms−x+10=1add terms−x=1−10subtract from both sides−x=−9add termsx=9negate both sides

(20)

LinearSolveSteps-x = 1+2, x;  

−x=1+2−x=3add termsx=−3negate both sides

(21)

LinearSolveSteps-x = 1+y, x; 

−x=1+yx=−1+ynegate both sidesx=−1−ydistribute negation

(22)

LinearSolveSteps5x/4+1/2 = 2x-1/2, x, implicitmultiply;  

5⁢x4+12=2⁢x−125⁢x4−2⁢x=−12−12subtract from both sides−3⁢x4=−12−12add terms−3⁢x4=−1add terms−3⁢x=4⁢−1multiply rhs by denominator of lhs−3⁢x=−4multiply constantsx=−4−3divide both sidesx=43reduce fraction by gcd

(23)

LinearSolveSteps.35y - .2 = .15y + .1, y, implicitmultiply;

0.35⁢y−0.2=0.15⁢y+0.10.35⁢y−0.15⁢y=0.1+0.2subtract from both sides0.20⁢y=0.1+0.2add terms0.20⁢y=0.3add termsy=0.30.20divide both sidesy=1.500000000divide constants

(24)

LinearSolveSteps4x-1 = 4(x+3), x, implicitmultiply;

4⁢x−1=4⁢x+34⁢x−1=4⁢x+4·3distributive multiply4⁢x−1=4⁢x+12multiply constants4⁢x−4⁢x=12+1subtract from both sides0=12+1add terms0=13add terms0=13no solution

(25)

LinearSolveSteps5x+10 = 5(x+2), x, implicitmultiply;

5⁢x+10=5⁢x+25⁢x+10=5⁢x+5·2distributive multiply5⁢x+10=5⁢x+10multiply constants5⁢x−5⁢x=10−10subtract from both sides0=10−10add terms0=0add terms0=0infinite number of solutions

(26)

LinearSolveStepsxy + 6x = 1, x, implicitmultiply;

x⁢y+6⁢x=1x⁢6+y=1factorx=16+ydivide both sides

(27)

LinearSolveSteps(x+1)/(2*y*z) = 4*y^2/z + 3*x/y, x; 

x+12⁢y⁢z=4⁢y2z+3⁢xyx+12⁢y⁢z−3⁢xy=4⁢y2zsubtract from both sidesy⁢x+12⁢y⁢z⁢y+2⁢y⁢z⁢−3⁢x2⁢y⁢z⁢y=4⁢y2zfind common denominatory⁢x+1+2⁢y⁢z⁢−3⁢x2⁢y⁢z⁢y=4⁢y2zsum over common denominatory⁢x+y·1+2⁢y⁢z⁢−3⁢x2⁢y⁢z⁢y=4⁢y2zdistributive multiplyx⁢y+y+−6⁢y⁢z⁢x2⁢y⁢z⁢y=4⁢y2zmultiply constants−6⁢x⁢y⁢z+x⁢y+y2⁢y⁢z⁢y=4⁢y2zreorder termsy⁢−6⁢x⁢z+x+1y·2⁢y⁢z=4⁢y2zfactor−6⁢x⁢z+x+12⁢y⁢z=4⁢y2zdivide−6⁢x⁢z+x+1=2⁢y⁢z⁢4⁢y2zmultiply rhs by denominator of lhs−6⁢x⁢z+x=2⁢y⁢z⁢4⁢y2z−1subtract from both sides−6⁢x⁢z+x=8⁢y3⁢zz−1multiply fraction−6⁢x⁢z+x=8⁢y3−1dividex⁢1−6⁢z=8⁢y3−1factorx=8⁢y3−11−6⁢zdivide both sides

(28)

LinearSolveSteps1/x = 3/4, x;  

1x=34x=43reciprocal of both sides

(29)

LinearSolveSteps1/x = 4, x;  

1x=4x=14reciprocal of both sides

(30)

LinearSolveSteps1/x - 1/2 = 3/4 - 2/x, x;  

1x−12=34−2x1x+2x=34+12subtract from both sides3x=34+12add terms3x=54add termsx3=45reciprocal of both sidesx=4513divide both sidesx=45⁢31rewrite division as multiplication by reciprocalx=125multiply fraction and reduce by gcd

(31)

LinearSolveSteps3(n - 1.8) + 2(n-1) = 2(n + 1) - 3(n-2), n, implicitmultiply;

3⁢n−1.8+2⁢n−1=2⁢n+1−3⁢n−23⁢n−1.8+2⁢n−1−2⁢n+1+3⁢n−2=0subtract from both sides3⁢n+3⁢−1.8+2⁢n−1−2⁢n+1+3⁢n−2=0distributive multiply3⁢n−5.4+2⁢n−1−2⁢n+1+3⁢n−2=0multiply constants3⁢n−5.4+2⁢n+2⁢−1−2⁢n+1+3⁢n−2=0distributive multiply3⁢n−5.4+2⁢n−2−2⁢n+1+3⁢n−2=0multiply constants3⁢n−5.4+2⁢n−2−2⁢n+2·1+3⁢n−2=0distributive multiply3⁢n−5.4+2⁢n−2−2⁢n+2+3⁢n−2=0multiply constants3⁢n−5.4+2⁢n−2−2⁢n+2+3⁢n+3⁢−2=0distributive multiply3⁢n−5.4+2⁢n−2−2⁢n+2+3⁢n−6=0multiply constants6⁢n−15.4=0add terms6⁢n=15.4subtract from both sidesn=15.46divide both sidesn=2.566666667divide constants

(32)

LinearSolveSteps3(n - 1.8) + n*(2-1) = 2n + 1, n, implicitmultiply;

3⁢n−1.8+n⁢2−1=2⁢n+13⁢n−1.8+n⁢2−1−2⁢n=1subtract from both sides3⁢n+3⁢−1.8+n⁢2−1−2⁢n=1distributive multiply3⁢n−5.4+n⁢2−1−2⁢n=1multiply constants3⁢n−5.4+n·1−2⁢n=1add terms2⁢n−5.4=1add terms2⁢n=1+5.4subtract from both sides2⁢n=6.4add termsn=6.42divide both sidesn=3.200000000divide constants

(33)

LinearSolveSteps5x/4+1/2 = 2x-1/2, x, implicitmultiply; 

5⁢x4+12=2⁢x−125⁢x4−2⁢x=−12−12subtract from both sides−3⁢x4=−12−12add terms−3⁢x4=−1add terms−3⁢x=4⁢−1multiply rhs by denominator of lhs−3⁢x=−4multiply constantsx=−4−3divide both sidesx=43reduce fraction by gcd

(34)

LinearSolveStepsx*(1-6*z) = 8*y-1, x;

x⁢1−6⁢z=8⁢y−1x=8⁢y−11−6⁢zdivide both sides

(35)

LinearSolveSteps3*(x-6*z) = 8*y-1, x;  

3⁢x−6⁢z=8⁢y−13⁢x+3⁢−6⁢z=8⁢y−1distributive multiply3⁢x+−18⁢z=8⁢y−1multiply constants3⁢x=8⁢y−1−−18⁢zsubtract from both sides3⁢x=8⁢y−1+18⁢zdistribute negationx=8⁢y−1+18⁢z3divide both sides

(36)

LinearSolveSteps10-3x = 7+2x, x, implicitmultiply;

10−3⁢x=7+2⁢x−3⁢x−2⁢x=7−10subtract from both sides−5⁢x=7−10add terms−5⁢x=−3add termsx=−3−5divide both sidesx=35reduce fraction by gcd

(37)

LinearSolveSteps10-3x = 7, x, implicitmultiply; 

10−3⁢x=7−3⁢x=7−10subtract from both sides−3⁢x=−3add termsx=−3−3divide both sidesx=1divide out common terms

(38)

LinearSolveSteps10−3⋅x = 7+3⋅x−y/4z, x; 

10+−3⁢x=7+3⁢x+−14⁢yz−3⁢x−7+3⁢x+−14⁢yz=−10subtract from both sides−3⁢x−7+3⁢x+−y4z=−10multiply fractionz⁢−3⁢xz+−7+3⁢x+−y4z=−10find common denominatorz⁢−3⁢x−7+3⁢x+−y4z=−10sum over common denominator−3⁢x⁢z−7−3⁢x+y4z=−10distribute negation−3⁢x⁢z−7−3⁢x+y4=z⁢−10multiply rhs by denominator of lhs−3⁢x⁢z−3⁢x=z⁢−10+7−y4subtract from both sides−3⁢x⁢z−3⁢x=7−10⁢z−y4reorder termsx⁢−3−3⁢z=7−10⁢z−y4factorx=7−10⁢z−y4−3−3⁢zdivide both sides

(39)

LinearSolveSteps1x+2=1,x;

x+2−1=1x+2=1reciprocal of both sidesx=1−2subtract from both sidesx=−1add terms

(40)

LinearSolveStepsxx+2=1,x;

xx+2=1x=x+2·1multiply rhs by denominator of lhsx=x+2multiply by 1x−x=2subtract from both sides0=2add terms0=2no solution

(41)

ExpandSteps(3*a)*(4*a-y+42);

3⁢a⁢4⁢a−y+42=3⁢a·4⁢a+3⁢a⁢−y+3⁢a·42distributive multiply=12⁢a⁢a+3⁢a⁢−y+3⁢a·42multiply constants=12⁢a2+3⁢a⁢−y+3⁢a·42multiply terms to exponential form=12⁢a2+−3⁢a⁢y+3⁢a·42multiply constants=12⁢a2−3⁢a⁢y+126⁢amultiply constants

(42)

ExpandSteps(3*a)*(4*a-y+42);    

3⁢a⁢4⁢a−y+42=3⁢a·4⁢a+3⁢a⁢−y+3⁢a·42distributive multiply=12⁢a⁢a+3⁢a⁢−y+3⁢a·42multiply constants=12⁢a2+3⁢a⁢−y+3⁢a·42multiply terms to exponential form=12⁢a2+−3⁢a⁢y+3⁢a·42multiply constants=12⁢a2−3⁢a⁢y+126⁢amultiply constants

(43)

ExpandSteps(x^2)*(x^3);

x2⁢x3=x5add exponents with common base

(44)

ExpandSteps(x^2*y/(x*y)); 

x2⁢yx⁢y=x2xdivide out common terms=xdivide

(45)

ExpandSteps(2*x^2*y/(4*x*y));

2⁢x2⁢y4⁢x⁢y=2⁢x24⁢xdivide out common terms=2⁢x4divide out common terms=x2reduce fraction by gcd

(46)

ExpandSteps(2.1*x)/4.3; 

2.1⁢x4.3=0.4883720930⁢xdivide constants

(47)

ExpandSteps(2.1*x^2*y/(4.3*x*y));

2.1⁢x2⁢y4.3⁢x⁢y=2.1⁢x24.3⁢xdivide out common terms=2.100000000⁢x4.3divide out common terms=0.4883720930⁢xdivide constants

(48)

ExpandSteps(x^2*y+y^2⋅x)/(x+y);

x2⁢y+y2⁢xx+y=x+y⁢x⁢yx+yfactor=x⁢ydivide

(49)

ExpandSteps(-y)^2;

−y2=−12⁢y2distribute exponent to individual terms=1⁢y2evaluate power

(50)

ExpandSteps(-y^2)+y^2;

−y2+y2=0add terms

(51)

ExpandSteps(x^2-y^2)/(x+y);

x2−y2x+y=x+y⁢x−yx+yfactor=x−ydivide

(52)

ExpandSteps2*(-y^2);

2⁢−y2=−2⁢y2multiply constants

(53)

ExpandSteps2*(x^2-y^2);

2⁢x2−y2=2⁢x2+2⁢−y2distributive multiply=2⁢x2+−2⁢y2multiply constants

(54)

ExpandSteps(2*(x^2-y^2))/(4*(x+y));

2⁢x2−y24⁢x+y=2⁢x2+2⁢−y24⁢x+ydistributive multiply=2⁢x2+−2⁢y24⁢x+ymultiply constants=2⁢x2−2⁢y24⁢x+4⁢ydistributive multiply=2⁢x+2⁢y⁢x−y2⁢x+2⁢y·2factor=x−y2divide

(55)

ExpandSteps(2.1*(x^2-y^2))/4;

2.1⁢x2−y24=2.1⁢x2+2.1⁢−y24distributive multiply=2.1⁢x2+−2.1⁢y24multiply constants=0.5250000000⁢x2−0.5250000000⁢y2divide constants

(56)

ExpandSteps(2.1*(x^2-y^2))/(4*(x+y));

2.1⁢x2−y24⁢x+y=2.1⁢x2+2.1⁢−y24⁢x+ydistributive multiply=2.1⁢x2+−2.1⁢y24⁢x+ymultiply constants=2.1⁢x2−2.1⁢y24⁢x+4⁢ydistributive multiply=x+y⁢2.100000000⁢x−2.100000000⁢yx+y·4.factor=2.100000000⁢x−2.100000000⁢y4.divide=0.5250000000⁢x−0.5250000000⁢ydivide constants

(57)

ExpandSteps(x^2/z)*(z^3/x^2);

x2z⁢z3x2=x2⁢z3z⁢x2multiply fraction=x2⁢z2x2divide out common terms=z2divide out common terms

(58)

ExpandSteps(17*x^4*y^2/(64*z^5)) * (24*y*z^2/(85*x^2));

17⁢x4⁢y264⁢z5⁢24⁢y⁢z285⁢x2=408⁢x4⁢y3⁢z25440⁢z5⁢x2multiply fraction=408⁢x4⁢y35440⁢x2⁢z3divide out common terms=408⁢x2⁢y35440⁢z3divide out common terms=3⁢x2⁢y340⁢z3reduce fraction by gcd

(59)

ExpandSteps3^2;

32=9evaluate power

(60)

ExpandSteps`%+``%+`9*a^2,6*a*b,`%+`6*a*b,4*b^2;

9⁢a2+6⁢a⁢b+6⁢a⁢b+4⁢b2=9⁢a2+12⁢a⁢b+4⁢b2add terms

(61)

ExpandSteps(3*a+2*b)^2;

3⁢a+2⁢b2=3⁢a+2⁢b⁢3⁢a+2⁢brewrite exponentiation as multiplication=3⁢a⁢3⁢a+2⁢b+2⁢b⁢3⁢a+2⁢bdistributive multiply=3⁢a·3⁢a+3⁢a·2⁢b+2⁢b⁢3⁢a+2⁢bdistributive multiply=9⁢a⁢a+3⁢a·2⁢b+2⁢b⁢3⁢a+2⁢bmultiply constants=9⁢a2+3⁢a·2⁢b+2⁢b⁢3⁢a+2⁢bmultiply terms to exponential form=9⁢a2+6⁢a⁢b+2⁢b⁢3⁢a+2⁢bmultiply constants=9⁢a2+6⁢a⁢b+2⁢b·3⁢a+2⁢b·2⁢bdistributive multiply=9⁢a2+6⁢a⁢b+6⁢b⁢a+2⁢b·2⁢bmultiply constants=9⁢a2+6⁢a⁢b+6⁢a⁢b+4⁢b⁢bmultiply constants=9⁢a2+6⁢a⁢b+6⁢a⁢b+4⁢b2multiply terms to exponential form=9⁢a2+12⁢a⁢b+4⁢b2add terms

(62)

ExpandSteps(3a+2b)*(4a-y+42), implicitmultiply;

3⁢a+2⁢b⁢4⁢a−y+42=3⁢a⁢4⁢a−y+42+2⁢b⁢4⁢a−y+42distributive multiply=3⁢a·4⁢a+3⁢a⁢−y+3⁢a·42+2⁢b⁢4⁢a−y+42distributive multiply=12⁢a⁢a+3⁢a⁢−y+3⁢a·42+2⁢b⁢4⁢a−y+42multiply constants=12⁢a2+3⁢a⁢−y+3⁢a·42+2⁢b⁢4⁢a−y+42multiply terms to exponential form=12⁢a2+−3⁢a⁢y+3⁢a·42+2⁢b⁢4⁢a−y+42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+2⁢b⁢4⁢a−y+42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+2⁢b·4⁢a+2⁢b⁢−y+2⁢b·42distributive multiply=12⁢a2−3⁢a⁢y+126⁢a+8⁢b⁢a+2⁢b⁢−y+2⁢b·42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+8⁢a⁢b+−2⁢b⁢y+2⁢b·42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+8⁢a⁢b−2⁢b⁢y+84⁢bmultiply constants=12⁢a2+8⁢a⁢b−3⁢a⁢y−2⁢b⁢y+126⁢a+84⁢breorder terms

(63)

ExpandSteps3*3;

3·3=9multiply constants

(64)

ExpandSteps1*2*3*4*5*6*7*8*9;

1·2·3·4·5·6·7·8·9=362880multiply constants

(65)

ExpandSteps1+1;

1+1=2add terms

(66)

ExpandSteps0^x;

0x=0evaluate power

(67)

ExpandStepsx^0;

x0=1x^0 = 1

(68)

ExpandSteps5^0;

50=1x^0 = 1

(69)

ExpandSteps(a*b)^3;

a⁢b3=a3⁢b3distribute exponent to individual terms

(70)

ExpandStepsa^3*a^2;

a3⁢a2=a5add exponents with common base

(71)

ExpandSteps(a+b)^2;

a+b2=a+b⁢a+brewrite exponentiation as multiplication=a⁢a+b+b⁢a+bdistributive multiply=a⁢a+a⁢b+b⁢a+bdistributive multiply=a2+a⁢b+b⁢a+bmultiply terms to exponential form=a2+a⁢b+b⁢a+b⁢bdistributive multiply=a2+a⁢b+a⁢b+b2multiply terms to exponential form=a2+2⁢a⁢b+b2add terms

(72)

ExpandStepsa+b5;

a+b5=a+b⁢a+b⁢a+b⁢a+b⁢a+brewrite exponentiation as multiplication=a⁢a+b+b⁢a+b⁢a+b⁢a+b⁢a+bdistributive multiply=a⁢a+a⁢b+b⁢a+b⁢a+b⁢a+b⁢a+bdistributive multiply=a2+a⁢b+b⁢a+b⁢a+b⁢a+b⁢a+bmultiply terms to exponential form=a2+a⁢b+b⁢a+b⁢b⁢a+b⁢a+b⁢a+bdistributive multiply=a2+a⁢b+a⁢b+b2⁢a+b⁢a+b⁢a+bmultiply terms to exponential form=a2+2⁢a⁢b+b2⁢a+b⁢a+b⁢a+badd terms=a2+2⁢a⁢b+b2⁢a+a2+2⁢a⁢b+b2⁢b⁢a+b⁢a+bdistributive multiply=a⁢a2+a·2⁢a⁢b+a⁢b2+a2+2⁢a⁢b+b2⁢b⁢a+b⁢a+bdistributive multiply=a3+a·2⁢a⁢b+a⁢b2+a2+2⁢a⁢b+b2⁢b⁢a+b⁢a+badd exponents with common base=a3+2⁢a2⁢b+a⁢b2+a2+2⁢a⁢b+b2⁢b⁢a+b⁢a+bmultiply terms to exponential form=a3+2⁢a2⁢b+a⁢b2+b⁢a2+b·2⁢a⁢b+b⁢b2⁢a+b⁢a+bdistributive multiply=a3+2⁢a2⁢b+a⁢b2+a2⁢b+2⁢b2⁢a+b⁢b2⁢a+b⁢a+bmultiply terms to exponential form=a3+2⁢a2⁢b+a⁢b2+a2⁢b+2⁢a⁢b2+b3⁢a+b⁢a+badd exponents with common base=a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢a+b⁢a+badd terms=a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢a+a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢b⁢a+bdistributive multiply=a⁢a3+a·3⁢a2⁢b+a·3⁢a⁢b2+a⁢b3+a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢b⁢a+bdistributive multiply=a4+a·3⁢a2⁢b+a·3⁢a⁢b2+a⁢b3+a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢b⁢a+badd exponents with common base=a4+3⁢a3⁢b+a·3⁢a⁢b2+a⁢b3+a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢b⁢a+badd exponents with common base=a4+3⁢a3⁢b+3⁢a2⁢b2+a⁢b3+a3+3⁢a2⁢b+3⁢a⁢b2+b3⁢b⁢a+bmultiply terms to exponential form=a4+3⁢a3⁢b+3⁢a2⁢b2+a⁢b3+b⁢a3+b·3⁢a2⁢b+b·3⁢a⁢b2+b⁢b3⁢a+bdistributive multiply=a4+3⁢a3⁢b+3⁢a2⁢b2+a⁢b3+a3⁢b+3⁢b2⁢a2+b·3⁢a⁢b2+b⁢b3⁢a+bmultiply terms to exponential form=a4+3⁢a3⁢b+3⁢a2⁢b2+a⁢b3+a3⁢b+3⁢a2⁢b2+3⁢b3⁢a+b⁢b3⁢a+badd exponents with common base=a4+3⁢a3⁢b+3⁢a2⁢b2+a⁢b3+a3⁢b+3⁢a2⁢b2+3⁢a⁢b3+b4⁢a+badd exponents with common base=a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢a+badd terms=a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢a+a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢bdistributive multiply=a⁢a4+a·4⁢a3⁢b+a·6⁢a2⁢b2+a·4⁢a⁢b3+a⁢b4+a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢bdistributive multiply=a5+a·4⁢a3⁢b+a·6⁢a2⁢b2+a·4⁢a⁢b3+a⁢b4+a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢badd exponents with common base=a5+4⁢a4⁢b+a·6⁢a2⁢b2+a·4⁢a⁢b3+a⁢b4+a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢badd exponents with common base=a5+4⁢a4⁢b+6⁢a3⁢b2+a·4⁢a⁢b3+a⁢b4+a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢badd exponents with common base=a5+4⁢a4⁢b+6⁢a3⁢b2+4⁢a2⁢b3+a⁢b4+a4+4⁢a3⁢b+6⁢a2⁢b2+4⁢a⁢b3+b4⁢bmultiply terms to exponential form=a5+4⁢a4⁢b+6⁢a3⁢b2+4⁢a2⁢b3+a⁢b4+b⁢a4+b·4⁢a3⁢b+b·6⁢a2⁢b2+b·4⁢a⁢b3+b⁢b4distributive multiply=a5+4⁢a4⁢b+6⁢a3⁢b2+4⁢a2⁢b3+a⁢b4+a4⁢b+4⁢b2⁢a3+b·6⁢a2⁢b2+b·4⁢a⁢b3+b⁢b4multiply terms to exponential form=a5+4⁢a4⁢b+6⁢a3⁢b2+4⁢a2⁢b3+a⁢b4+a4⁢b+4⁢a3⁢b2+6⁢b3⁢a2+b·4⁢a⁢b3+b⁢b4add exponents with common base=a5+4⁢a4⁢b+6⁢a3⁢b2+4⁢a2⁢b3+a⁢b4+a4⁢b+4⁢a3⁢b2+6⁢a2⁢b3+4⁢b4⁢a+b⁢b4add exponents with common base=a5+4⁢a4⁢b+6⁢a3⁢b2+4⁢a2⁢b3+a⁢b4+a4⁢b+4⁢a3⁢b2+6⁢a2⁢b3+4⁢a⁢b4+b5add exponents with common base=a5+5⁢a4⁢b+10⁢a3⁢b2+10⁢a2⁢b3+5⁢a⁢b4+b5add terms

(73)

Note that this could be expanded but the system chooses not to as the output would be excessively large (the cut-off is an exponent ≥ 100)

  ExpandStepsa+b1000; 

a+b1000

(74)

ExpandSteps(a+b)^(1/2) ⋅ (a+b)^(3/2);

a+b12⁢a+b32=a+b2add exponents with common base=a+b⁢a+brewrite exponentiation as multiplication=a⁢a+b+b⁢a+bdistributive multiply=a⁢a+a⁢b+b⁢a+bdistributive multiply=a2+a⁢b+b⁢a+bmultiply terms to exponential form=a2+a⁢b+b⁢a+b⁢bdistributive multiply=a2+a⁢b+a⁢b+b2multiply terms to exponential form=a2+2⁢a⁢b+b2add terms

(75)

ExpandSteps(1+I)^1.5;

1+I1.5=1+I1.5add terms=0.6435942529+1.553773974·Ievaluate power

(76)

ExpandStepsa/(2*b/a);

a2⁢ba=a⁢a2⁢brewrite division as multiplication by reciprocal=a22⁢bmultiply fraction and reduce by gcd

(77)

ExpandStepsa/(2*a);

a2⁢a=12divide out common terms

(78)

ExpandSteps3a/(6a), implicitmultiply;

3⁢a6⁢a=36divide out common terms=12reduce fraction by gcd

(79)

ExpandSteps(3x sin(x))/x, implicitmultiply; 

3⁢x⁢sin⁡xx=3⁢sin⁡xdivide out common terms

(80)

ExpandSteps3( sin(x) + y ), implicitmultiply;

3⁢sin⁡x+y=3⁢sin⁡x+3⁢ydistributive multiply

(81)

ExpandSteps(3a+2b)*(4a-y^2+42), implicitmultiply;

3⁢a+2⁢b⁢4⁢a−y2+42=3⁢a+2⁢b⁢−y2+4⁢a+42reorder terms=−y2+4⁢a+42·3⁢a+−y2+4⁢a+42·2⁢bdistributive multiply=3⁢a⁢−y2+3⁢a·4⁢a+3⁢a·42+−y2+4⁢a+42·2⁢bdistributive multiply=−3⁢a⁢y2+3⁢a·4⁢a+3⁢a·42+−y2+4⁢a+42·2⁢bmultiply constants=−3⁢a⁢y2+12⁢a⁢a+3⁢a·42+−y2+4⁢a+42·2⁢bmultiply constants=−3⁢a⁢y2+12⁢a2+3⁢a·42+−y2+4⁢a+42·2⁢bmultiply terms to exponential form=−3⁢a⁢y2+12⁢a2+126⁢a+−y2+4⁢a+42·2⁢bmultiply constants=−3⁢a⁢y2+12⁢a2+126⁢a+2⁢b⁢−y2+2⁢b·4⁢a+2⁢b·42distributive multiply=−3⁢a⁢y2+12⁢a2+126⁢a+−2⁢b⁢y2+2⁢b·4⁢a+2⁢b·42multiply constants=−3⁢a⁢y2+12⁢a2+126⁢a−2⁢b⁢y2+8⁢b⁢a+2⁢b·42multiply constants=−3⁢a⁢y2+12⁢a2+126⁢a−2⁢b⁢y2+8⁢a⁢b+84⁢bmultiply constants=−3⁢a⁢y2−2⁢b⁢y2+12⁢a2+8⁢a⁢b+126⁢a+84⁢breorder terms

(82)

ExpandSteps3(a^2-1)/(a+1), implicitmultiply;

3⁢a2−1a+1=3⁢a2+3⁢−1a+1distributive multiply=3⁢a2−3a+1multiply constants=a+1⁢3⁢a−3a+1factor=3⁢a−3divide

(83)

ExpandStepsa-b-c^2-1/2-a/2-(a^2-4)^2; 

a−b−c2−12−a2−a2−42=a−b−c2−12−a2−a2−4⁢a2−4rewrite exponentiation as multiplication=a−b−c2−12−a2−a2⁢a2−4+−4⁢a2−4distributive multiply=a−b−c2−12−a2−a2⁢a2+a2⁢−4+−4⁢a2−4distributive multiply=a−b−c2−12−a2−a4+a2⁢−4+−4⁢a2−4add exponents with common base=a−b−c2−12−a2−a4−4⁢a2+−4⁢a2+−4⁢−4distributive multiply=a−b−c2−12−a2−a4−4⁢a2−4⁢a2+16multiply constants=a−b−c2−12−a2−a4+8⁢a2−16add terms=−332+a2−b−c2−a4+8⁢a2add terms

(84)

ExpandStepsx^2 y / (xy) , implicitmultiply;

x2⁢yx⁢y=x2xdivide out common terms=xdivide

(85)

ExpandSteps2x^2*y/(4xy), implicitmultiply;

2⁢x2⁢y4⁢x⁢y=2⁢x⁢y4⁢ydivide out common terms=2⁢x4divide out common terms=x2reduce fraction by gcd

(86)

ExpandSteps`%*``%*`3,a,42;

3⁢a·42=126⁢amultiply constants

(87)

ExpandSteps3a * (4a-y+42), implicitmultiply;

3⁢a⁢4⁢a−y+42=3⁢a·4⁢a+3⁢a⁢−y+3⁢a·42distributive multiply=12⁢a⁢a+3⁢a⁢−y+3⁢a·42multiply constants=12⁢a2+3⁢a⁢−y+3⁢a·42multiply terms to exponential form=12⁢a2+−3⁢a⁢y+3⁢a·42multiply constants=12⁢a2−3⁢a⁢y+126⁢amultiply constants

(88)

ExpandSteps(3*a+2*b)^3;

3⁢a+2⁢b3=3⁢a+2⁢b⁢3⁢a+2⁢b⁢3⁢a+2⁢brewrite exponentiation as multiplication=3⁢a⁢3⁢a+2⁢b+2⁢b⁢3⁢a+2⁢b⁢3⁢a+2⁢bdistributive multiply=3⁢a·3⁢a+3⁢a·2⁢b+2⁢b⁢3⁢a+2⁢b⁢3⁢a+2⁢bdistributive multiply=9⁢a⁢a+3⁢a·2⁢b+2⁢b⁢3⁢a+2⁢b⁢3⁢a+2⁢bmultiply constants=9⁢a2+3⁢a·2⁢b+2⁢b⁢3⁢a+2⁢b⁢3⁢a+2⁢bmultiply terms to exponential form=9⁢a2+6⁢a⁢b+2⁢b⁢3⁢a+2⁢b⁢3⁢a+2⁢bmultiply constants=9⁢a2+6⁢a⁢b+2⁢b·3⁢a+2⁢b·2⁢b⁢3⁢a+2⁢bdistributive multiply=9⁢a2+6⁢a⁢b+6⁢b⁢a+2⁢b·2⁢b⁢3⁢a+2⁢bmultiply constants=9⁢a2+6⁢a⁢b+6⁢a⁢b+4⁢b⁢b⁢3⁢a+2⁢bmultiply constants=9⁢a2+6⁢a⁢b+6⁢a⁢b+4⁢b2⁢3⁢a+2⁢bmultiply terms to exponential form=9⁢a2+12⁢a⁢b+4⁢b2⁢3⁢a+2⁢badd terms=9⁢a2+12⁢a⁢b+4⁢b2·3⁢a+9⁢a2+12⁢a⁢b+4⁢b2·2⁢bdistributive multiply=3⁢a·9⁢a2+3⁢a·12⁢a⁢b+3⁢a·4⁢b2+9⁢a2+12⁢a⁢b+4⁢b2·2⁢bdistributive multiply=27⁢a⁢a2+3⁢a·12⁢a⁢b+3⁢a·4⁢b2+9⁢a2+12⁢a⁢b+4⁢b2·2⁢bmultiply constants=27⁢a3+3⁢a·12⁢a⁢b+3⁢a·4⁢b2+9⁢a2+12⁢a⁢b+4⁢b2·2⁢badd exponents with common base=27⁢a3+36⁢a⁢a⁢b+3⁢a·4⁢b2+9⁢a2+12⁢a⁢b+4⁢b2·2⁢bmultiply constants=27⁢a3+36⁢a2⁢b+3⁢a·4⁢b2+9⁢a2+12⁢a⁢b+4⁢b2·2⁢bmultiply terms to exponential form=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+9⁢a2+12⁢a⁢b+4⁢b2·2⁢bmultiply constants=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+2⁢b·9⁢a2+2⁢b·12⁢a⁢b+2⁢b·4⁢b2distributive multiply=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+18⁢b⁢a2+2⁢b·12⁢a⁢b+2⁢b·4⁢b2multiply constants=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+18⁢a2⁢b+24⁢b⁢a⁢b+2⁢b·4⁢b2multiply constants=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+18⁢a2⁢b+24⁢b2⁢a+2⁢b·4⁢b2multiply terms to exponential form=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+18⁢a2⁢b+24⁢a⁢b2+8⁢b⁢b2multiply constants=27⁢a3+36⁢a2⁢b+12⁢a⁢b2+18⁢a2⁢b+24⁢a⁢b2+8⁢b3add exponents with common base=27⁢a3+54⁢a2⁢b+36⁢a⁢b2+8⁢b3add terms

(89)

ExpandSteps(3*a+2*b)*(4*a-y+42);

3⁢a+2⁢b⁢4⁢a−y+42=3⁢a⁢4⁢a−y+42+2⁢b⁢4⁢a−y+42distributive multiply=3⁢a·4⁢a+3⁢a⁢−y+3⁢a·42+2⁢b⁢4⁢a−y+42distributive multiply=12⁢a⁢a+3⁢a⁢−y+3⁢a·42+2⁢b⁢4⁢a−y+42multiply constants=12⁢a2+3⁢a⁢−y+3⁢a·42+2⁢b⁢4⁢a−y+42multiply terms to exponential form=12⁢a2+−3⁢a⁢y+3⁢a·42+2⁢b⁢4⁢a−y+42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+2⁢b⁢4⁢a−y+42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+2⁢b·4⁢a+2⁢b⁢−y+2⁢b·42distributive multiply=12⁢a2−3⁢a⁢y+126⁢a+8⁢b⁢a+2⁢b⁢−y+2⁢b·42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+8⁢a⁢b+−2⁢b⁢y+2⁢b·42multiply constants=12⁢a2−3⁢a⁢y+126⁢a+8⁢a⁢b−2⁢b⁢y+84⁢bmultiply constants=12⁢a2+8⁢a⁢b−3⁢a⁢y−2⁢b⁢y+126⁢a+84⁢breorder terms

(90)

ExpandSteps(3*a)*(4*a-y+42); 

3⁢a⁢4⁢a−y+42=3⁢a·4⁢a+3⁢a⁢−y+3⁢a·42distributive multiply=12⁢a⁢a+3⁢a⁢−y+3⁢a·42multiply constants=12⁢a2+3⁢a⁢−y+3⁢a·42multiply terms to exponential form=12⁢a2+−3⁢a⁢y+3⁢a·42multiply constants=12⁢a2−3⁢a⁢y+126⁢amultiply constants

(91)

ExpandSteps3(x-2), implicitmultiply;

3⁢x−2=3⁢x+3⁢−2distributive multiply=3⁢x−6multiply constants

(92)

ExpandSteps-(1+y);

−1+y=−1−ydistribute negation

(93)

ExpandSteps-(1+y+y^2); 

−1+y+y2=−y2−y−1distribute negation

(94)

ExpandStepsy⋅((3+1)/(2*y) - 3*x/y); 

y⁢3+12⁢y−3⁢xy=y⁢42⁢y−3⁢xyadd terms=y⁢2y−3⁢xyreduce fraction by gcd=y⁢2y+y⁢−3⁢xydistributive multiply=2⁢yy+y⁢−3⁢xymultiply fraction=2+y⁢−3⁢xydivide=2+−3⁢x⁢yymultiply fraction=2−3⁢xdivide

(95)

ExpandSteps-3x -2x, implicitmultiply;

−3⁢x−2⁢x=−5⁢xadd terms

(96)

ExpandSteps7+3⋅x−y/4z; 

7+3⁢x+−14⁢yz=7+3⁢x+−y4zmultiply fraction

(97)

 

In this example, the input is not quoted so some automatic simplifications happen before ExpandSteps sees the input.

  ExpandSteps17⋅x4⋅y264⋅z5+24⋅y⋅z285⋅x2⋅x2z⋅z3x2;

1764⁢x4⁢y2⁢z−5+2485⁢y⁢z4⁢x−2=17⁢x4⁢y264⁢z5+2485⁢y⁢z4⁢x−2multiply fraction=17⁢x4⁢y264⁢z5+24⁢y⁢z485⁢x2multiply fraction

(98)

ExpandSteps17⋅x4⋅y264⋅z5⋅24⋅y⋅z285⋅x2⋅x2z⋅z3x2;

340⁢x2⁢y3z=3⁢x2⁢y340zmultiply fraction=3⁢x2⁢y340⁢zreduce fraction by gcd

(99)

ExpandSteps"((17*x^4*y^2)/(64*z^5)) * (24⋅y⋅z^2/85⋅x^2  * x^2/z * z^3/x^2");

17⁢x4⁢y264⁢z5⁢24⁢y⁢z285⁢x2⁢x2z⁢z3x2=408⁢x6⁢y3⁢z55440⁢z6⁢x4multiply fraction=408⁢x6⁢y35440⁢z⁢x4divide out common terms=408⁢x2⁢y35440⁢zdivide out common terms=3⁢x2⁢y340⁢zreduce fraction by gcd

(100)