Learn on PengiCalifornia Reveal Math, Algebra 1Unit 7: Exponents and Roots

7-2 Operations with Negative Exponents

In this Grade 9 California Reveal Math Algebra 1 lesson, students learn to simplify expressions containing zero and negative exponents using the Zero Exponent Property, the Negative Exponent Property, and the Quotient of Powers Property. The lesson covers rewriting expressions with negative exponents as equivalent fractions with positive exponents and applying these rules to multi-variable algebraic expressions. Students also explore the concept of order of magnitude as a real-world application of powers of ten.

Section 1

Exploring Zero and Negative Exponents

Property

As the exponent of a base decreases by 1, the value of the power is divided by the base. Following this pattern past the exponent of 1 reveals the rules for zero and negative exponents:
For every nonzero number $x$ , $x^0 = 1$ .
For every nonzero number $x$ , $x^{-n} = \frac{1}{x^n}$ .

Examples

Consider the pattern for powers of 2:

2^3 = 8

2^2 = 4 \quad (\text{since } 8 \div 2 = 4)

2^1 = 2 \quad (\text{since } 4 \div 2 = 2)

2^0 = 1 \quad (\text{since } 2 \div 2 = 1)

2^{-1} = \frac{1}{2} \quad (\text{since } 1 \div 2 = \frac{1}{2})

2^{-2} = \frac{1}{4} \quad (\text{since } \frac{1}{2} \div 2 = \frac{1}{4})

Notice that $2^{-2}$ is exactly the same as $\frac{1}{2^2}$ .

Simplify $\frac{x^{-4}}{y^2}$ : Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in $\frac{1}{x^4 y^2}$ .

Explanation

Zero and negative exponents aren't magic; they are just the logical continuation of a mathematical pattern! Every time an exponent drops by one, you divide by the base. This proves why any non-zero number to the power of zero is exactly 1. It also shows that a negative exponent is basically a "flip-it" command: it tells you to take the reciprocal of the base and make the exponent positive.

Section 2

Negative and Zero Exponents

Property

Zero as an Exponent

a^0 = 1, \quad \text{if } a \neq 0

Negative Exponents

a^{-n} = \frac{1}{a^n} \quad \text{if } a \neq 0

A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative. The laws of exponents apply to negative exponents.

Section 3

Zero and Negative Exponents

Property

Zero Exponent Property
If $a$ is a non-zero number, then $a^0 = 1$ .

Properties of Negative Exponents
If $n$ is an integer and $a \neq 0$ , then $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$ .

Quotient to a Negative Power Property
If $a$ and $b$ are real numbers, $a \neq 0$ , $b \neq 0$ and $n$ is an integer, then

(\frac{a}{b})^{-n} = (\frac{b}{a})^n

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Exploring Zero and Negative Exponents

Property

Examples

Consider the pattern for powers of 2:

2^3 = 8

2^2 = 4 \quad (\text{since } 8 \div 2 = 4)

2^1 = 2 \quad (\text{since } 4 \div 2 = 2)

2^0 = 1 \quad (\text{since } 2 \div 2 = 1)

2^{-1} = \frac{1}{2} \quad (\text{since } 1 \div 2 = \frac{1}{2})

2^{-2} = \frac{1}{4} \quad (\text{since } \frac{1}{2} \div 2 = \frac{1}{4})

Notice that $2^{-2}$ is exactly the same as $\frac{1}{2^2}$ .

Simplify $\frac{x^{-4}}{y^2}$ : Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in $\frac{1}{x^4 y^2}$ .

Explanation

Section 2

Negative and Zero Exponents

Property

Zero as an Exponent

a^0 = 1, \quad \text{if } a \neq 0

Negative Exponents

a^{-n} = \frac{1}{a^n} \quad \text{if } a \neq 0

A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative. The laws of exponents apply to negative exponents.

Section 3

Zero and Negative Exponents

Property

Zero Exponent Property
If $a$ is a non-zero number, then $a^0 = 1$ .

Properties of Negative Exponents
If $n$ is an integer and $a \neq 0$ , then $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$ .

Quotient to a Negative Power Property
If $a$ and $b$ are real numbers, $a \neq 0$ , $b \neq 0$ and $n$ is an integer, then

(\frac{a}{b})^{-n} = (\frac{b}{a})^n

Overview

Lesson overview

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

Overview

Section 1

Exploring Zero and Negative Exponents

Property

Examples

Consider the pattern for powers of 2:

2^3 = 8

2^2 = 4 \quad (\text{since } 8 \div 2 = 4)

2^1 = 2 \quad (\text{since } 4 \div 2 = 2)

2^0 = 1 \quad (\text{since } 2 \div 2 = 1)

2^{-1} = \frac{1}{2} \quad (\text{since } 1 \div 2 = \frac{1}{2})

2^{-2} = \frac{1}{4} \quad (\text{since } \frac{1}{2} \div 2 = \frac{1}{4})

Notice that $2^{-2}$ is exactly the same as $\frac{1}{2^2}$ .

Simplify $\frac{x^{-4}}{y^2}$ : Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in $\frac{1}{x^4 y^2}$ .

Explanation

Section 2

Negative and Zero Exponents

Property

Zero as an Exponent

a^0 = 1, \quad \text{if } a \neq 0

Negative Exponents

a^{-n} = \frac{1}{a^n} \quad \text{if } a \neq 0

A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative. The laws of exponents apply to negative exponents.

Section 3

Zero and Negative Exponents

Property

Zero Exponent Property
If $a$ is a non-zero number, then $a^0 = 1$ .

Properties of Negative Exponents
If $n$ is an integer and $a \neq 0$ , then $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$ .

Quotient to a Negative Power Property
If $a$ and $b$ are real numbers, $a \neq 0$ , $b \neq 0$ and $n$ is an integer, then

(\frac{a}{b})^{-n} = (\frac{b}{a})^n