Preliminary Notions

Distributive Property

$$a(b + c) = ab + ac$$

$$(+) \cdot (+) = (+)$$

$$(-) \cdot (+) = (-)$$

$$(+) \cdot (-) = (-)$$

$$(-) \cdot (-) = (+)$$

Associative Property

$$a \cdot (b \cdot c) = (a \cdot b) \cdot c$$

$$a^{m} \cdot a^{n} = a^{m+n}$$

$$(a \cdot b)^{n} = a^{n} \cdot b^{n}$$

$$(a^m)^n = a^{m \cdot n}$$

$$(a^{\alpha} \cdot b^{\beta})^n = a^{\alpha n} \cdot b^{\beta n}$$

$$\frac{(+)}{(+)} = (+)$$

$$\frac{(-)}{(+)} = (-)$$

$$\frac{(+)}{(-)} = (-)$$

$$\frac{(-)}{(-)} = (+)$$

$$\frac{a^{m}}{a^{n}} = a^{m-n} \quad (a \neq 0)$$

$$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$$

$$\left( \frac{a^\alpha}{b^\beta} \right)^n = \frac{a^{\alpha n}}{b^{\beta n}} \quad (b \neq 0)$$

Inverses

$$a^{-n} = \frac{1}{a^n} \quad \text{; } a \neq 0$$

$0^{-n}$ is undefined for $n > 0$.

Distributive Property

$$\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c} \quad (c \neq 0)$$

Condition: $y, z, w, k \neq 0$

$$\frac{x}{y} = x \left( \frac{1}{y} \right)$$

$$\left( \frac{x}{y} \right) \left( \frac{w}{k} \right) = \frac{xw}{yk}$$

$$\frac{xy}{wx} = \frac{y}{w}$$

$$\frac{x}{y} + \frac{z}{y} = \frac{x + z}{y}$$

$$\frac{x}{y} + \frac{w}{z} = \frac{xz + yw}{yz}$$

$$\frac{x}{y} \div \frac{w}{z} = \frac{xz}{yw}$$

$$x + \frac{y}{w} = \frac{xw + y}{w}$$

Division by zero is undefined. All denominators must be $\neq 0$.