## Working with Exponents

The following are a list of many of the rules associated with exponents. I've found this an incredibly useful cheatsheet for helping perform simplifications.

*Also make sure to remember your order of operations when working with exponents*

**Product**

Rule:

$${x^r} \cdot {x^2} = X^{r+s}$$

Example:

$$x^3 \cdot x^5 = x^8$$

**Quotient**

Rule:

$$\frac{x^r}{x^s} = x^{r-s}$$

Example:

$$\frac{a^6}{a^2} = 2^{4}$$

**Power of a Power**

Rule:

$$\left( x^r\right) ^s = x^{rs}$$

Example:

$$\left( z^2\right) ^5 = z^{10}$$

**Power of a Product**

Rule:

$$\left( x \cdot y\right) ^r = x^r \cdot y^r$$

Example:

$$\left( a \cdot b\right) ^2 = a^2 \cdot b^2$$

**Power of a Quotient**

Rule:

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

Example:

$$\left( \frac{5}{7}\right) ^3 = \frac{5^3}{7^3}$$

**Negative Exponent**

Rule:

$$x^{-r} = \frac{1}{x^r}$$

Example:

$$b^{-6} = \frac{1}{b^6}$$

**Zero Exponent**

Rule:

$$x^0 = 1, \textrm{if } x \neq 0$$

Example:

$$7^0 = 1$$

**Negative Exponent with Quotient**

Rule:

$$\left(\frac{x}{y}\right)^{-r} = \left(\frac{y}{x}\right)^r$$

Example:

$$\left( \frac{1}{2}\right) ^{-3} = \left( \frac{2}{1}\right) ^3 = 2^3$$

$$5^{-2} = \left( \frac{5}{1}\right) ^{-2} = \left( \frac{1}{5}\right) ^2$$

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