# Binomial Theorem/Examples

## Examples of Use of Binomial Theorem

### Cube of Sum

$\paren {x + y}^3 = x^3 + 3 x^2 y + 3 x y^2 + y^3$

### Cube of Difference

$\paren {x - y}^3 = x^3 - 3 x^2 y + 3 x y^2 - y^3$

### Fourth Power of Sum

$\paren {x + y}^4 = x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4$

### Fourth Power of Difference

$\paren {x - y}^4 = x^4 - 4 x^3 y + 6 x^2 y^2 - 4 x y^3 + y^4$

### Fifth Power of Sum

$\paren {x + y}^5 = x^5 + 5 x^4 y + 10 x^3 y^2 + 10 x^2 y^3 + 5 x y^4 + y^5$

### Fifth Power of Difference

$\paren {x - y}^5 = x^5 - 5 x^4 y + 10 x^3 y^2 - 10 x^2 y^3 + 5 x y^4 - y^5$

### Sixth Power of Sum

$\paren {x + y}^6 = x^6 + 6 x^5 y + 15 x^4 y^2 + 20 x^3 y^3 + 15 x^2 y^4 + 6 x y^5 + y^6$

### Sixth Power of Difference

$\paren {x - y}^6 = x^6 - 6 x^5 y + 15 x^4 y^2 - 20 x^3 y^3 + 15 x^2 y^4 - 6 x y^5 + y^6$

### Power of $11$: $11^4$

$11^4 = \left({10 + 1}\right)^4 = 14 \, 641$

### Binomial Theorem: $\paren {1 + x}^7$

$\paren {1 + x}^7 = 1 + 7 x + 21 x^2 + 35 x^3 + 35 x^4 + 21 x^5 + 7 x^6 + x^7$

### Square Root of 2

$\sqrt 2 = 2 \paren {1 - \dfrac 1 {2^2} - \dfrac 1 {2^5} - \dfrac 1 {2^7} - \dfrac 5 {2^{11} } - \cdots}$