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1. Square of a binomial:

\[

(x + y)^2 = x^2 + 2xy + y^2

\]

2. Square of a binomial (with subtraction):

\[

(x – y)^2 = x^2 – 2xy + y^2

\]

3. Cube of a binomial:

\[

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

\]

4. Cube of a binomial (with subtraction):

\[

(x – y)^3 = x^3 – 3x^2y + 3xy^2 – y^3

\]

5. Fourth power of a binomial:

\[

(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

\]

6. Fourth power of a binomial (with subtraction):

\[

(x – y)^4 = x^4 – 4x^3y + 6x^2y^2 – 4xy^3 + y^4

\]

7. Fifth power of a binomial:

\[

(x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

\]

8. Fifth power of a binomial (with subtraction):

\[

(x – y)^5 = x^5 – 5x^4y + 10x^3y^2 – 10x^2y^3 + 5xy^4 – y^5

\]

9. Sixth power of a binomial:

\[

(x + y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6

\]

10. Sixth power of a binomial (with subtraction):

\[

(x – y)^6 = x^6 – 6x^5y + 15x^4y^2 – 20x^3y^3 + 15x^2y^4 – 6xy^5 + y^6

\]

1. Difference of squares:

\[

x^2 – y^2 = (x – y)(x + y)

\]

2. Difference of cubes:

\[

x^3 – y^3 = (x – y)(x^2 + xy + y^2)

\]

3. Sum of cubes:

\[

x^3 + y^3 = (x + y)(x^2 – xy + y^2)

\]

4. Difference of fourth powers:

\[

x^4 – y^4 = (x – y)(x + y)(x^2 + y^2)

\]

5. Difference of fifth powers:

\[

x^5 – y^5 = (x – y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)

\]

6. Sum of fifth powers:

\[

x^5 + y^5 = (x + y)(x^4 – x^3y + x^2y^2 – xy^3 + y^4)

\]

7. Difference of sixth powers:

\[

x^6 – y^6 = (x – y)(x + y)(x^2 + y^2)(x^2 – xy + y^2)

\]