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1.
Sum of fifth powers
$a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$
2.
Square of difference
$(a-b)^2=a^2-2ab+b^2$
3.
Square of sum
$(a+b)^2=a^2+2ab+b^2$
4.
Cube of difference
$(a-b)^3=a^3-3a^2b+3ab^2-b^3$
5.
Cube of sum
$(a+b)^3=a^3+3a^2b+3ab^2+b^3$
6.
Fourth power of difference
$(a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4$
7.
Fourth power of sum
$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$
8.
Sum of cubes
$a^3+b^3=(a+b)(a^2-ab+b^2)$
9.
Difference of squares
$a^2-b^2=(a-b)(a+b)$