Lesson 1: Integer Exponents

New Concept

This lesson expands exponents to include integers. You'll learn the meaning of negative and zero exponents, like $a^{-n} = \frac{1}{a^n}$, and apply all exponent laws to simplify expressions and model real-world phenomena.

What’s next

First, you’ll work through examples of simplifying with negative exponents. Then, you'll apply these skills in practice cards on power functions and scientific notation.

Property

Definition of Negative and Zero Exponents.

Property

Definition of Power Function.

A function of the form

where $k$ and $p$ are nonzero constants, is called a power function. For example, inverse variation functions like $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x^2}$ can be written as power functions, $f(x) = x^{-1}$ and $g(x) = x^{-2}$.

Examples

• The function $f(x) = \frac{10}{x^4}$ is a power function because it can be rewritten as $f(x) = 10x^{-4}$, where $k=10$ and $p=-4$.
• The function $g(x) = 2x^3 + 5$ is not a power function because it includes the addition of a constant term, 5.
• The function $h(x) = \frac{3}{4x^2}$ is a power function that can be written as $h(x) = \frac{3}{4}x^{-2}$, with $k = \frac{3}{4}$ and $p=-2$.

Property

Laws of Exponents.
The laws are valid for all integer exponents $m$ and $n$, and for $a, b \neq 0$.

I $a^m \cdot a^n = a^{m+n}$

II $\frac{a^m}{a^n} = a^{m-n}$

Property

To Write a Number in Scientific Notation.

1. Locate the decimal point so that there is exactly one nonzero digit to its left.

2. Count the number of places you moved the decimal point: this determines the power of 10.