# The formula for abbreviated multiplication

The most important abbreviated multiplication formulas.

Formulas of abbreviated multiplication allow you to perform calculations much faster.
The most commonly used abbreviated multiplication formulas:

#### a3 − b3 = (a − b)(a2 + ab + b2)

The abbreviated multiplication formulas are helpful for multiplying or expanding algebraic expressions. They facilitate efficient counting. There are a lot of these patterns. We list a few below, which are used most often.

The square of the sum of the numbers

• (a + b)2 = a2 + 2ab + b2
e.g: 312 = (30+1)2 = 302+2×30+1 = 900+60+1 = 961
• does not occur equality: (a+b)2 = a2 + b2
e.g 25 = (3+2)2 32 + 22 = 13
• justification of the formula by the bill:
(a + b)2 = (a + b) × (a + b) =
Aa + ab + Ba + Bb = a2 + 2ab + b2

The square of the difference of the numbers

• (a b)2 = a2 – 2ab + b2
e.g: 292 = (30-1)2 = 302-2×30+1 = 900-60+1 = 841
• does not occur equality: (a-b)2 = a2b2
e.g 1 = (3-2)2 32 – 22 = 5
• justification of the formula:
(a – b)2 = (a – b) × (a – b) = Aa ab Ba + Bb = a2 – 2ab + b2

A square of the sum of three numbers

• (a+b+c)2 = a2 + b2 + c2 + 2ab + 2Ac + 2Bc
e.g: 1112 = (100+10+1)2 = 1002 + 102 +1 +2×100×10 + 2×100 + 2×10 = 10000 + 100 + 1 + 2000 + 200 + 20 = 12321
• does not occur equality: (A+B+c)2 = a2 + b2 + c2
e.g 36 = (3+2+1)2 32 + 22 + 12 = 14
• justification of the formula:
(a + b + c)2 = (a + b + c) × (a + b + c) = Aa + ab + Ac + Ba + Bb + Bc + ca + Cb + cc = a2 + b2 + c2 + 2ab + 2Ac + 2Bc

Product of the sum and difference of numbers = Difference of squares of numbers

• (a + b)×(ab) = a2 b2
e.g: 101×99 = (100+1)×(100-1) = 1002 – 1 = 9999
• justification of the formula :
(a + b) × (a – b) = Aa ab + Ba Bb = a2 b2

A cube of the sum of numbers

• (a + b)3 = a3 + 3a2b + 3ab2 + b3
e.g: 1013 = (100+1)3 = 1003 + 3×1002 + 3×100 + 1 =
= 1000000 + 30000 + 300 + 1 = 1030301
• does not occur equality: (a+b)3 = a3 + b3
e.g 125 = (3+2)3 33 + 23 = 35
• justification of the formula by the bill:
(a + b)3 = (a + b) × (a + b) × (a + b)
= (Aa + ab + Ba + Bb) × (a + b) = Aaa + aab + aba + fig + Baa + Bab + Bba + bbb =
= a3 + 3a2b + 3ab2 + b3

Cube of number difference

• (a b)3 = a3 – 3a2b + 3ab2 b3
e.g: 993 = (100-1)3 = 1003 – 3×1002 + 3×100 – 1 =
= 1000000 – 30000 + 300 – 1 = 970299

Sum of cubes of numbers

a3 + b3 = (a + b)×(a2 ab + b2)

justification of the formula:

(a + b)×(a2 ab + b2) = Aa2 aab + ab2 + Ba2 Bab + Bb2= a3a2b + ab2 + a2bab2 + b3 =
= a3 + b3

The difference of the cubes of numbers

a3 b3 = (ab)×(a2 + ab + b2)

justification of the formula:

(ab)×(a2 + ab + b2) = Aa2 + aab + ab2 Ba2 BabBb2 = a3 + a2b + ab2 a2bab2 b3 =
= a3b3

Difference of fourth powers of numbers

a4 b4 = (ab)×(a3 + a2b + ab2 + b3) = (a + b)×(a3a2b + ab2b3)

Addition n-these powers of numbers (For n odd!!!)

an + bn = (a + b) (an-1an-2b + an-3b2 – … + bn-1)

Difference n-these powers of numbers (For n even!!!)

anbn = (a + b) (an-1an-2b + an-3b2 – … + bn-1)

Difference n-these powers of numbers (for everyone n natural)

anbn = (a b) (an-1 + an-2b + an-3b2 + … + a2bn-3 + abn-2 + bn-1)