Learn on PengiBig Ideas Math, Course 3Chapter 10: Exponents and Scientific Notation

Lesson 4: Zero and Negative Exponents

In this Grade 8 lesson from Big Ideas Math Course 3, students learn how to evaluate expressions with zero exponents using the rule a⁰ = 1 and with negative integer exponents using the definition a⁻ⁿ = 1/aⁿ. Students apply the Product of Powers and Quotient of Powers Properties to simplify expressions and rewrite them using only positive exponents. The lesson also connects these concepts to real-world rate problems and place value patterns.

Section 1

Zero as an Exponent

Property

$a^0 = 1$ , if $a \neq 0$

This definition is based on the second law of exponents. For any non-zero number $a$ , the quotient $\frac{a^n}{a^n}$ is equal to 1. Using the law of exponents, we can also write $\frac{a^n}{a^n} = a^{n-n} = a^0$ . Therefore, it is logical to define $a^0$ as 1.

Examples

For a positive integer, $8^0 = 1$ .
For a negative integer, $(-55)^0 = 1$ .
For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$ .

Section 2

Multiplicative Inverse Property and Negative Exponents

Property

The multiplicative inverse (reciprocal) of $a^n$ is $a^{-n}$ , and vice versa: $a^n \cdot a^{-n} = a^0 = 1$

a^n \cdot a^{-n} = 1 \text{ where } a \neq 0

Section 3

Simplifying Algebraic Expressions with Zero Exponents

Property

When simplifying algebraic expressions containing zero exponents, replace any term with exponent zero with 1: $a^0 = 1$ for any nonzero number $a$ .

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Zero as an Exponent

Property

$a^0 = 1$ , if $a \neq 0$

Examples

For a positive integer, $8^0 = 1$ .
For a negative integer, $(-55)^0 = 1$ .
For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$ .

Section 2

Multiplicative Inverse Property and Negative Exponents

Property

The multiplicative inverse (reciprocal) of $a^n$ is $a^{-n}$ , and vice versa: $a^n \cdot a^{-n} = a^0 = 1$

a^n \cdot a^{-n} = 1 \text{ where } a \neq 0

Section 3

Simplifying Algebraic Expressions with Zero Exponents

Property

When simplifying algebraic expressions containing zero exponents, replace any term with exponent zero with 1: $a^0 = 1$ for any nonzero number $a$ .

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Zero as an Exponent

Property

$a^0 = 1$ , if $a \neq 0$

Examples

For a positive integer, $8^0 = 1$ .
For a negative integer, $(-55)^0 = 1$ .
For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$ .

Section 2

Multiplicative Inverse Property and Negative Exponents

Property

The multiplicative inverse (reciprocal) of $a^n$ is $a^{-n}$ , and vice versa: $a^n \cdot a^{-n} = a^0 = 1$

a^n \cdot a^{-n} = 1 \text{ where } a \neq 0

Section 3

Simplifying Algebraic Expressions with Zero Exponents

Property

When simplifying algebraic expressions containing zero exponents, replace any term with exponent zero with 1: $a^0 = 1$ for any nonzero number $a$ .