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.