Formulas of abridged multiplication

From the rules of multiplication of sums and polynomials the following seven formulas of abridged multiplication can be easily received. It is necessary to know them by heart, as they are used in most of problems in mathematics.

[1] ( a + b ) ² =  a ² + 2ab + b ² ,

[2] ( a – b ) ² =  a ² – 2ab + b ² ,

[3] ( a + b ) ( a – b ) = a ² – b ² ,

[4] ( a + b ) ³ =  a ³ + 3a ² b + 3ab ² + b ³ ,

[5] ( a – b ) ³ =  a ³ – 3a ² b + 3ab ² – b ³ ,

[6] ( a + b )( a ² – ab + b ² ) =  a ³ + b ³ ,

[7] ( a – b )( a ² + ab + b ² ) =  a ³ – b ³ .

E x a m p l e :  Calculate  99³  using the formula [5] .

S o l u t i o n :  99³ = (100 – 1)³ = 1000000 – 3 ·10000 ·1 + 3 ·100 ·1 – 1 = 970299.