Mathematical tables: short multiplication formulas
Tables shows various short multiplication formulas. Both common formulas such as (a + b)² (square of a sum) and general cases (e.g. any power of a sum) are presented.

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Short multiplication formulas

NameFormula
Square of sumShow source(a+b)2=a2+2 a b+b2{\left( a+\mathrm{b}\right)}^{2}={ a}^{2}+2~ a~\mathrm{b}+{\mathrm{b}}^{2}
Square of differenceShow source(ab)2=a22 a b+b2{\left( a-\mathrm{b}\right)}^{2}={ a}^{2}-2~ a~\mathrm{b}+{\mathrm{b}}^{2}
Cube of a sumShow source(a+b)3=a3+3 a2 b+3 a b2+b3{\left( a+\mathrm{b}\right)}^{3}={ a}^{3}+3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}
Cube of a differenceShow source(ab)3=a33 a2 b+3 a b2b3{\left( a-\mathrm{b}\right)}^{3}={ a}^{3}-3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}-{\mathrm{b}}^{3}
Sum of a cubesShow sourcea3+b3=(a+b) (a2a b+b2){ a}^{3}+{\mathrm{b}}^{3}=\left( a+\mathrm{b}\right)~\left({ a}^{2}- a~\mathrm{b}+{\mathrm{b}}^{2}\right)
Difference of a cubsShow sourcea3b3=(ab) (a2+a b+b2){ a}^{3}-{\mathrm{b}}^{3}=\left( a-\mathrm{b}\right)~\left({ a}^{2}+ a~\mathrm{b}+{\mathrm{b}}^{2}\right)
Sophie Germain identityShow sourcea4+4 b4=(a2+2 a b+2 b2) (a22 a b+2 b2){ a}^{4}+4~{\mathrm{b}}^{4}=\left({ a}^{2}+2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)~\left({ a}^{2}-2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)
Difference of a fourth powersShow sourcea4b4=(ab) (a3+a2 b+a b2+b3){ a}^{4}-{\mathrm{b}}^{4}=\left( a-\mathrm{b}\right)~\left({ a}^{3}+{ a}^{2}~\mathrm{b}+ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}\right)
Sum of a fifth powersShow sourcea5+b5=(a+b) (a4a3 b+a2 b2a b3+b4){ a}^{5}+{\mathrm{b}}^{5}=\left( a+\mathrm{b}\right)~\left({ a}^{4}-{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}- a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)
Difference of a fifth powersShow sourcea5b5=(ab) (a4+a3 b+a2 b2+a b3+b4){ a}^{5}-{\mathrm{b}}^{5}=\left( a-\mathrm{b}\right)~\left({ a}^{4}+{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}+ a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)
Square of three terms: (a + b + c)²Show source(a+b+c)2=a2+b2+c2+2 a b+2 a c+2 b c{\left( a+\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}+2~ a~ c+2~\mathrm{b}~ c
Square of three terms: (a + b - c)²Show source(a+bc)2=a2+b2+c2+2 a b2 a c2 b c{\left( a+\mathrm{b}- c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}-2~ a~ c-2~\mathrm{b}~ c
Square of three terms: (a - b + c)²Show source(ab+c)2=a2+b2+c22 a b+2 a c2 b c{\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}+2~ a~ c-2~\mathrm{b}~ c
Square of three terms: (a - b - c)²Show source(ab+c)2=a2+b2+c22 a b2 a c+2 b c{\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}-2~ a~ c+2~\mathrm{b}~ c
Any power of a sum - general formulaShow source(a+b)n=k=0n(nk)ankbk=k=0nn!k!(nk)!ankbk(a + b)^n = \sum_{k=0}^{n}{n \choose k} a^{n-k} b^k = \sum_{k=0}^{n}{\frac{n!}{k!(n-k)!}} a^{n-k} b^k
Any power of a difference - general formulaShow source(ab)n=k=0n(1)k(nk)ankbk=k=0n(1)kn!k!(nk)!ankbk(a - b)^n = \sum_{k=0}^{n}{(-1)^k {n \choose k}} a^{n-k} b^k = \sum_{k=0}^{n}{(-1)^k \frac{n!}{k!(n-k)!}} a^{n-k} b^k
Square of a sum of any number of termsShow source(i=1kai)2=i=1kj=1kaiaj\left(\sum_{i=1}^{k}{a_i}\right)^2 = \sum_{i=1}^{k} \sum_{j=1}^{k} a_i a_j

Short multiplication formulas: formulas with square of a numbers

NameFormula
Square of sumShow source(a+b)2=a2+2 a b+b2{\left( a+\mathrm{b}\right)}^{2}={ a}^{2}+2~ a~\mathrm{b}+{\mathrm{b}}^{2}
Square of differenceShow source(ab)2=a22 a b+b2{\left( a-\mathrm{b}\right)}^{2}={ a}^{2}-2~ a~\mathrm{b}+{\mathrm{b}}^{2}

Short multiplication formulas: formulas with cube of a numbers

NameFormula
Cube of a sumShow source(a+b)3=a3+3 a2 b+3 a b2+b3{\left( a+\mathrm{b}\right)}^{3}={ a}^{3}+3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}
Cube of a differenceShow source(ab)3=a33 a2 b+3 a b2b3{\left( a-\mathrm{b}\right)}^{3}={ a}^{3}-3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}-{\mathrm{b}}^{3}
Sum of a cubesShow sourcea3+b3=(a+b) (a2a b+b2){ a}^{3}+{\mathrm{b}}^{3}=\left( a+\mathrm{b}\right)~\left({ a}^{2}- a~\mathrm{b}+{\mathrm{b}}^{2}\right)
Difference of a cubsShow sourcea3b3=(ab) (a2+a b+b2){ a}^{3}-{\mathrm{b}}^{3}=\left( a-\mathrm{b}\right)~\left({ a}^{2}+ a~\mathrm{b}+{\mathrm{b}}^{2}\right)

Short multiplication formulas: formulas with higher powers

NameFormula
Sophie Germain identityShow sourcea4+4 b4=(a2+2 a b+2 b2) (a22 a b+2 b2){ a}^{4}+4~{\mathrm{b}}^{4}=\left({ a}^{2}+2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)~\left({ a}^{2}-2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)
Difference of a fourth powersShow sourcea4b4=(ab) (a3+a2 b+a b2+b3){ a}^{4}-{\mathrm{b}}^{4}=\left( a-\mathrm{b}\right)~\left({ a}^{3}+{ a}^{2}~\mathrm{b}+ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}\right)
Sum of a fifth powersShow sourcea5+b5=(a+b) (a4a3 b+a2 b2a b3+b4){ a}^{5}+{\mathrm{b}}^{5}=\left( a+\mathrm{b}\right)~\left({ a}^{4}-{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}- a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)
Difference of a fifth powersShow sourcea5b5=(ab) (a4+a3 b+a2 b2+a b3+b4){ a}^{5}-{\mathrm{b}}^{5}=\left( a-\mathrm{b}\right)~\left({ a}^{4}+{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}+ a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)

Short multiplication formulas: three terms

NameFormula
Square of three terms: (a + b + c)²Show source(a+b+c)2=a2+b2+c2+2 a b+2 a c+2 b c{\left( a+\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}+2~ a~ c+2~\mathrm{b}~ c
Square of three terms: (a + b - c)²Show source(a+bc)2=a2+b2+c2+2 a b2 a c2 b c{\left( a+\mathrm{b}- c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}-2~ a~ c-2~\mathrm{b}~ c
Square of three terms: (a - b + c)²Show source(ab+c)2=a2+b2+c22 a b+2 a c2 b c{\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}+2~ a~ c-2~\mathrm{b}~ c
Square of three terms: (a - b - c)²Show source(ab+c)2=a2+b2+c22 a b2 a c+2 b c{\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}-2~ a~ c+2~\mathrm{b}~ c

Short multiplication formulas: general formulas

NameFormula
Any power of a sum - general formulaShow source(a+b)n=k=0n(nk)ankbk=k=0nn!k!(nk)!ankbk(a + b)^n = \sum_{k=0}^{n}{n \choose k} a^{n-k} b^k = \sum_{k=0}^{n}{\frac{n!}{k!(n-k)!}} a^{n-k} b^k
Any power of a difference - general formulaShow source(ab)n=k=0n(1)k(nk)ankbk=k=0n(1)kn!k!(nk)!ankbk(a - b)^n = \sum_{k=0}^{n}{(-1)^k {n \choose k}} a^{n-k} b^k = \sum_{k=0}^{n}{(-1)^k \frac{n!}{k!(n-k)!}} a^{n-k} b^k
Square of a sum of any number of termsShow source(i=1kai)2=i=1kj=1kaiaj\left(\sum_{i=1}^{k}{a_i}\right)^2 = \sum_{i=1}^{k} \sum_{j=1}^{k} a_i a_j

Some facts

  • The short multiplication formulas allow quick performing of common mathematical operations e.g. a square of the sum of two numbers:
    ⓘ Example: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  • There is no obligation to use short multiplication formulas, because the same calculation can be done manually (step by step, multiplying all components one-by-one). However, the use of ready-made formulas may help to avoid tedious calculations and reduce the chance of making a mistake.
  • We can achieve the above formula by multiplying the terms one by one (colloquially: each by each):
    (a+b)2=(a+b)(a+b)=a(a+b)+b(a+b)=aa+ab+ba+bb=a2+2ab+b2 \begin{array}{l} (a + b)^2 = \\ (a + b)(a + b) = \\ a(a + b) + b(a + b) = \\ a \cdot a + ab + ba + b \cdot b = \\ a^2 + 2ab + b^2 \end{array}

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