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:

$$\left( \frac{1}{2}\right) ^{-3} = \left( \frac{2}{1}\right) ^3 = 2^3$$
$$5^{-2} = \left( \frac{5}{1}\right) ^{-2} = \left( \frac{1}{5}\right) ^2$$

If you liked what you read, subscribe to my newsletter and you will get my cheat sheet on Python, Machine Learning (ML), Natural Language Processing (NLP), SQL, and more. You will receive an email each time a new article is posted.

tags: math, school



Comments !