# Properties of Exponents

Posted by Brian Roepke on Tue 11 February 2020 Updated on Tue 11 February 2020

## 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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tags: math, school