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 $\left(a+\mathrm{b}\right)^{2}=a^{2}+2~a \cdot \mathrm{b}+\mathrm{b}^{2}$
Square of difference	Show source $\left(a-\mathrm{b}\right)^{2}=a^{2}-2~a \cdot \mathrm{b}+\mathrm{b}^{2}$
Cube of a sum	Show source $\left(a+\mathrm{b}\right)^{3}=a^{3}+3~a^{2} \cdot \mathrm{b}+3~a \cdot \mathrm{b}^{2}+\mathrm{b}^{3}$
Cube of a difference	Show source $\left(a-\mathrm{b}\right)^{3}=a^{3}-3~a^{2} \cdot \mathrm{b}+3~a \cdot \mathrm{b}^{2}-\mathrm{b}^{3}$
Sum of a cubes	Show source $a^{3}+\mathrm{b}^{3}=\left(a+\mathrm{b}\right) \cdot \left(a^{2}-a \cdot \mathrm{b}+\mathrm{b}^{2}\right)$
Difference of a cubs	Show source $a^{3}-\mathrm{b}^{3}=\left(a-\mathrm{b}\right) \cdot \left(a^{2}+a \cdot \mathrm{b}+\mathrm{b}^{2}\right)$
Sophie Germain identity	Show source $a^{4}+4~\mathrm{b}^{4}=\left(a^{2}+2~a \cdot \mathrm{b}+2~\mathrm{b}^{2}\right) \cdot \left(a^{2}-2~a \cdot \mathrm{b}+2~\mathrm{b}^{2}\right)$
Difference of a fourth powers	Show source $a^{4}-\mathrm{b}^{4}=\left(a-\mathrm{b}\right) \cdot \left(a^{3}+a^{2} \cdot \mathrm{b}+a \cdot \mathrm{b}^{2}+\mathrm{b}^{3}\right)$
Sum of a fifth powers	Show source $a^{5}+\mathrm{b}^{5}=\left(a+\mathrm{b}\right) \cdot \left(a^{4}-a^{3} \cdot \mathrm{b}+a^{2} \cdot \mathrm{b}^{2}-a \cdot \mathrm{b}^{3}+\mathrm{b}^{4}\right)$
Difference of a fifth powers	Show source $a^{5}-\mathrm{b}^{5}=\left(a-\mathrm{b}\right) \cdot \left(a^{4}+a^{3} \cdot \mathrm{b}+a^{2} \cdot \mathrm{b}^{2}+a \cdot \mathrm{b}^{3}+\mathrm{b}^{4}\right)$
Square of three terms: (a + b + c)²	Show source $\left(a+\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}+2~a \cdot \mathrm{b}+2~a \cdot c+2~\mathrm{b} \cdot c$
Square of three terms: (a + b - c)²	Show source $\left(a+\mathrm{b}-c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}+2~a \cdot \mathrm{b}-2~a \cdot c-2~\mathrm{b} \cdot c$
Square of three terms: (a - b + c)²	Show source $\left(a-\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}-2~a \cdot \mathrm{b}+2~a \cdot c-2~\mathrm{b} \cdot c$
Square of three terms: (a - b - c)²	Show source $\left(a-\mathrm{b}+c\right)^{2}=a^{2}+\mathrm{b}^{2}+c^{2}-2~a \cdot \mathrm{b}-2~a \cdot c+2~\mathrm{b} \cdot c$
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 $\left(a+\mathrm{b}\right)^{2}=a^{2}+2~a \cdot \mathrm{b}+\mathrm{b}^{2}$
Square of difference	Show source $\left(a-\mathrm{b}\right)^{2}=a^{2}-2~a \cdot \mathrm{b}+\mathrm{b}^{2}$

Short multiplication formulas: formulas with cube of a numbers#

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

Short multiplication formulas: formulas with higher powers#

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

Short multiplication formulas: three terms#

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

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):

$\begin{array}{l} (a + b)^2 = \\ (a + b)(a + b) = \\ a(a + b) + b(a + b) = \\ a \cdot a + ab + ba + b \cdot b = \\ a^2 + 2ab + b^2 \end{array}$

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