# Modeling, Functions, and Graphs

## Associative Laws.

If $$a\text{,}$$ $$b\text{,}$$ and $$c$$ are any numbers, then $$(a + b) + c = a + (b + c)\text{.}$$
Multiplication
If $$a\text{,}$$ $$b\text{,}$$ and $$c$$ are any numbers, then $$(a\cdot b)\cdot c = a\cdot (b\cdot c)\text{.}$$

## Commutative Laws.

If $$a$$ and $$b$$ are any numbers, then $$a + b = b + a\text{.}$$
Multiplication
If $$a$$ and $$b$$ are any numbers, then $$a\cdot b = b\cdot a\text{.}$$

## Distributive Law.

$$a(b + c) = ab + ac$$ for any numbers $$a\text{,}$$ $$b\text{,}$$ and $$c\text{.}$$

## Properties of Equality.

If $$a = b$$ and $$c$$ is any number, then $$a + c = b + c\text{.}$$
Subtraction:
If $$a = b$$ and $$c$$ is any number, then $$a - c = b - c\text{.}$$
Multiplication
If $$a = b$$ and $$c$$ is any number, then $$a\cdot c = b\cdot c\text{.}$$
Division
If $$a = b$$ and $$c$$ is any nonzero number, then $$\frac{a}{c} =\frac{b}{c} \text{.}$$

## Fundamental Principle of Fractions.

If $$a$$ is any number, and $$b$$ and $$c$$ are nonzero numbers, then $$\displaystyle{\frac{a\cdot c}{b\cdot c}= \frac{a}{b}}\text{.}$$

## Laws of Exponents.

1. $$\displaystyle a^m\cdot a^n = a^{m+n}$$
• $$\displaystyle \dfrac{a^m}{a^n}=a^{m-n} \hphantom{blank1}(n\lt m)$$
• $$\displaystyle \displaystyle{\frac{a^m}{a^n}=\frac{1}{a^{n-m}} \hphantom{blank}(n\gt m)}$$
2. $$\displaystyle \left(a^m\right)^n=a^{m+n}$$
3. $$\displaystyle (ab)^n=a^n b^n$$
4. $$\displaystyle \displaystyle{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} }$$

If $$a$$ and $$b$$ are both nonnegative, then $$\sqrt{ab}=\sqrt{a}\sqrt{b} \text{.}$$

If $$a\ge 0$$ and $$b\gt 0\text{,}$$ then $$\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}} \text{.}$$
If $$ab= 0$$ then either $$a= 0$$ or $$b=0 \text{.}$$