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#

 Name Formula Square of sum Show source${\left( a+\mathrm{b}\right)}^{2}={ a}^{2}+2~ a~\mathrm{b}+{\mathrm{b}}^{2}$ Square of difference Show source${\left( a-\mathrm{b}\right)}^{2}={ a}^{2}-2~ a~\mathrm{b}+{\mathrm{b}}^{2}$ Cube of a sum Show source${\left( a+\mathrm{b}\right)}^{3}={ a}^{3}+3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}$ Cube of a difference Show source${\left( a-\mathrm{b}\right)}^{3}={ a}^{3}-3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}-{\mathrm{b}}^{3}$ Sum of a cubes Show source${ a}^{3}+{\mathrm{b}}^{3}=\left( a+\mathrm{b}\right)~\left({ a}^{2}- a~\mathrm{b}+{\mathrm{b}}^{2}\right)$ Difference of a cubs Show source${ a}^{3}-{\mathrm{b}}^{3}=\left( a-\mathrm{b}\right)~\left({ a}^{2}+ a~\mathrm{b}+{\mathrm{b}}^{2}\right)$ Sophie Germain identity Show source${ a}^{4}+4~{\mathrm{b}}^{4}=\left({ a}^{2}+2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)~\left({ a}^{2}-2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)$ Difference of a fourth powers Show source${ a}^{4}-{\mathrm{b}}^{4}=\left( a-\mathrm{b}\right)~\left({ a}^{3}+{ a}^{2}~\mathrm{b}+ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}\right)$ Sum of a fifth powers Show source${ a}^{5}+{\mathrm{b}}^{5}=\left( a+\mathrm{b}\right)~\left({ a}^{4}-{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}- a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)$ Difference of a fifth powers Show source${ a}^{5}-{\mathrm{b}}^{5}=\left( a-\mathrm{b}\right)~\left({ a}^{4}+{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}+ a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)$ Square of three terms: (a + b + c)² Show source${\left( a+\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}+2~ a~ c+2~\mathrm{b}~ c$ Square of three terms: (a + b - c)² Show source${\left( a+\mathrm{b}- c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}-2~ a~ c-2~\mathrm{b}~ c$ Square of three terms: (a - b + c)² Show source${\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}+2~ a~ c-2~\mathrm{b}~ c$ Square of three terms: (a - b - c)² Show source${\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}-2~ a~ c+2~\mathrm{b}~ c$ Any power of a sum - general formula Show source$(a + b)^n = \sum_{k=0}^{n}{n \choose k} a^{n-k} b^k = \sum_{k=0}^{n}{\frac{n!}{k!(n-k)!}} a^{n-k} b^k$ Any power of a difference - general formula Show source$(a - b)^n = \sum_{k=0}^{n}{(-1)^k {n \choose k}} a^{n-k} b^k = \sum_{k=0}^{n}{(-1)^k \frac{n!}{k!(n-k)!}} a^{n-k} b^k$ Square of a sum of any number of terms Show source$\left(\sum_{i=1}^{k}{a_i}\right)^2 = \sum_{i=1}^{k} \sum_{j=1}^{k} a_i a_j$

# Short multiplication formulas: formulas with square of a numbers#

 Name Formula Square of sum Show source${\left( a+\mathrm{b}\right)}^{2}={ a}^{2}+2~ a~\mathrm{b}+{\mathrm{b}}^{2}$ Square of difference Show source${\left( a-\mathrm{b}\right)}^{2}={ a}^{2}-2~ a~\mathrm{b}+{\mathrm{b}}^{2}$

# Short multiplication formulas: formulas with cube of a numbers#

 Name Formula Cube of a sum Show source${\left( a+\mathrm{b}\right)}^{3}={ a}^{3}+3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}$ Cube of a difference Show source${\left( a-\mathrm{b}\right)}^{3}={ a}^{3}-3~{ a}^{2}~\mathrm{b}+3~ a~{\mathrm{b}}^{2}-{\mathrm{b}}^{3}$ Sum of a cubes Show source${ a}^{3}+{\mathrm{b}}^{3}=\left( a+\mathrm{b}\right)~\left({ a}^{2}- a~\mathrm{b}+{\mathrm{b}}^{2}\right)$ Difference of a cubs Show source${ a}^{3}-{\mathrm{b}}^{3}=\left( a-\mathrm{b}\right)~\left({ a}^{2}+ a~\mathrm{b}+{\mathrm{b}}^{2}\right)$

# Short multiplication formulas: formulas with higher powers#

 Name Formula Sophie Germain identity Show source${ a}^{4}+4~{\mathrm{b}}^{4}=\left({ a}^{2}+2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)~\left({ a}^{2}-2~ a~\mathrm{b}+2~{\mathrm{b}}^{2}\right)$ Difference of a fourth powers Show source${ a}^{4}-{\mathrm{b}}^{4}=\left( a-\mathrm{b}\right)~\left({ a}^{3}+{ a}^{2}~\mathrm{b}+ a~{\mathrm{b}}^{2}+{\mathrm{b}}^{3}\right)$ Sum of a fifth powers Show source${ a}^{5}+{\mathrm{b}}^{5}=\left( a+\mathrm{b}\right)~\left({ a}^{4}-{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}- a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)$ Difference of a fifth powers Show source${ a}^{5}-{\mathrm{b}}^{5}=\left( a-\mathrm{b}\right)~\left({ a}^{4}+{ a}^{3}~\mathrm{b}+{ a}^{2}~{\mathrm{b}}^{2}+ a~{\mathrm{b}}^{3}+{\mathrm{b}}^{4}\right)$

# Short multiplication formulas: three terms#

 Name Formula Square of three terms: (a + b + c)² Show source${\left( a+\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}+2~ a~ c+2~\mathrm{b}~ c$ Square of three terms: (a + b - c)² Show source${\left( a+\mathrm{b}- c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}+2~ a~\mathrm{b}-2~ a~ c-2~\mathrm{b}~ c$ Square of three terms: (a - b + c)² Show source${\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}+2~ a~ c-2~\mathrm{b}~ c$ Square of three terms: (a - b - c)² Show source${\left( a-\mathrm{b}+ c\right)}^{2}={ a}^{2}+{\mathrm{b}}^{2}+{ c}^{2}-2~ a~\mathrm{b}-2~ a~ c+2~\mathrm{b}~ c$

# Short multiplication formulas: general formulas#

 Name Formula Any power of a sum - general formula Show source$(a + b)^n = \sum_{k=0}^{n}{n \choose k} a^{n-k} b^k = \sum_{k=0}^{n}{\frac{n!}{k!(n-k)!}} a^{n-k} b^k$ Any power of a difference - general formula Show source$(a - b)^n = \sum_{k=0}^{n}{(-1)^k {n \choose k}} a^{n-k} b^k = \sum_{k=0}^{n}{(-1)^k \frac{n!}{k!(n-k)!}} a^{n-k} b^k$ Square of a sum of any number of terms Show source$\left(\sum_{i=1}^{k}{a_i}\right)^2 = \sum_{i=1}^{k} \sum_{j=1}^{k} a_i a_j$

# Some facts#

• The short multiplication formulas allow quick performing of common mathematical operations e.g. a square of the sum of two numbers:
ⓘ Example: $(a + b)^2 = a^2 + 2ab + b^2$
• 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):
$\begin{array}{l} (a + b)^2 = \\ (a + b)(a + b) = \\ a(a + b) + b(a + b) = \\ a \cdot a + ab + ba + b \cdot b = \\ a^2 + 2ab + b^2 \end{array}$