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## LarsWiki

calumus meretrix et gladio innocentis

### Math Formulas

#### Laws of Exponents

Generally these can be relied upon only for real numbers and positive bases (x and y). Exponents (n and m) can be irrational, but what some of these formulas mean in that case is not clear. {$0^0$} is indeterminate, but is defined as 1 in some restricted contexts. These are actually theorems and many of them can be derived from a relatively small subset. Some obvious extensions have been omitted.

{$$x^0 = 1$$}

{$$x^{-n} = {1 \over x^n}$$}

{$$x^n = \prod^n x$$}

{$$x^n x^m = x^{n+m}$$}

{$$(xy)^n = x^n y^n$$}

{$$(x^n)^m = x^{nm}$$}

{$$\left( x \over y \right)^n = {x^n \over y^n}$$}

{$$\sqrt[n] {x} = x^{ \frac {1} {n}}$$}

#### Binomial Coefficients

{$${n \choose k} = \frac{n!}{k!\,(n-k)!} = \prod_{\ell=1}^k \frac{n-\ell+1}{\ell}$$}

#### Binomial Theorem

{$$(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}$$}

#### Factors of Cubic Expressions

##### Difference of Two Cubes

{$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$}

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### December 23, 2018

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