# Introduction to Exponents

## What does it mean?

### Definitions:

$$x^n = x * x * x * x * x * x$$ note: There are n x's in the product.

x ≠ 0, x = base, n = exponent

### Rules:

Any number raised to the zero power (except 0) equals 1. $$x^0 = 1$$ Any number raised to the power of one equals itself. $$x^1 = x$$ To multiply terms with the same base, add the exponents. $$x^a * x^b = x^{(a+b)}$$ To divide terms with the same base, subtract the exponents. $$\frac{x^a}{x^b} = x^{(a-b)}$$ When a product has an exponent, each factor is raised to that power. $${(x^a)^b} = x^{a * b}$$ A number with a negative exponent equals its reciprocal with a positive exponent. $$x^{(-a)} = \frac{1}{x^a}$$ When a product of two numbers has an exponent, each factor is raised to that power. $${(x * y)^a} = x^a * y^a$$ An absolute value of the squared number equals squared value of the number. $$|x^2| = |x|^2 = x^2$$

## What does it look like?

A general example to help you recognize patterns and spot the information you're looking for

$$4^3 = 4 * 4 * 4$$ $$x^2 = x * x$$ $$5^0 = 1$$ $$7^1 = 7$$ $$3^2 * 3^4 = 3^{(2+4)} = 3^6 = 729$$ $$\frac{2^5}{2^3} = 2^{(5-3)} = 2^2 = 4$$ $$(3^2)^5 = 3^{(2*5)} = 3^{10} = 59049$$ $$10^{(-2)} = \frac{1}{10^2} = 0.01$$ $${(2*5)}^3 = 2^3 * 5^3 = 8 * 125 = 1000$$ $$|3^{(2)}| = |3|^2 = 3^2 = 9$$ $$|{(-3)}^{(2)}| = |{(-3)}|^2 = 3^2 = 9$$ Please remember: $${(-5)}^2 ≠ {-5}^2$$ $$(-5)^2 = {(-1)}^2 * 5^2 = 1 * 25 = 25$$ $${-5}^2 = {(-1)} * {(5)}^2 = {-1} * 25 = {-25}$$

## You'll use it...

They're used to describe large quantities like acid rain, growth of bacteria, strength of radiation, "pH" level of the water, earthquakes (the Richter scale), how loud sound is(the "decibel" level), and how bright stars All these things are described using exponents, and the laws of exponents are used to determine, say, how much stronger one earthquake is than another.

## Video

### Basic Exponents

Watch a Khan Academy Video »
Length: 3:02

## Video

### Negative Exponents

Watch a Khan Academy Video »
Length: 7:13