Learn on PengienVision, Mathematics, Grade 8Chapter 1: Real Numbers

Lesson 6: Use Properties of Integer Exponents

In this Grade 8 lesson from enVision Mathematics Chapter 1, students learn to apply four properties of integer exponents — the Product of Powers, Power of Products, Power of a Power, and Quotient of Powers properties — to write equivalent exponential expressions. Students practice multiplying and dividing expressions with the same base, raising a power to a power, and multiplying expressions with the same exponent but different bases. These skills build fluency in simplifying and rewriting exponential expressions using rules such as adding, subtracting, or multiplying exponents.

Section 1

Product and Quotient of Powers Properties

Property

When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$ .
When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$ .

Examples

Product Property: Multiply $10^4 \cdot 10^2$ .

$10^4 \cdot 10^2 = 10^{4+2} = 10^6$ .

Quotient Property: Divide $\frac{10^7}{10^3}$ .

$\frac{10^7}{10^3} = 10^{7-3} = 10^4$ .

Using Negative Exponents: Multiply $x^{-5} \cdot x^2$ .

$x^{-5} \cdot x^2 = x^{-5+2} = x^{-3}$ , which can be written as $\frac{1}{x^3}$ .

Explanation

These exponent properties act as mathematical shortcuts because multiplication is just repeated addition, and division is repeated subtraction. When you multiply powers of the same base, you are combining groups of factors, so you add the exponents. When you divide, you are canceling out groups of factors, so you subtract the exponents.

Section 2

Power of a Power Property

Property

To raise a power to a power, keep the same base and multiply the exponents. In symbols,

(a^m)^n = a^{mn}

Examples

To simplify $(x^3)^5$ , you multiply the exponents: $x^{3 \cdot 5} = x^{15}$ .
To simplify $(4^2)^3$ , you keep the base and multiply the powers: $4^{2 \cdot 3} = 4^6$ .
Be careful to distinguish from products: $(a^5)(a^2) = a^{5+2} = a^7$ , but $(a^5)^2 = a^{5 \cdot 2} = a^{10}$ .

Explanation

Think of this as repeated multiplication. $(x^4)^3$ is just $x^4$ multiplied by itself three times. Adding the exponents $4+4+4$ is the same as multiplying $4 \cdot 3$ . So, you multiply the exponents.

Section 3

Power of a Product Property

Property

To raise a product to a power, raise each factor to the power. In symbols,

(ab)^n = a^nb^n

Examples

To simplify $(4xy)^2$ , apply the exponent to each factor inside: $4^2x^2y^2 = 16x^2y^2$ .
For $(-3a^2)^3$ , raise each factor to the third power: $(-3)^3(a^2)^3 = -27a^6$ .
Note the difference: in $5x^3$ , only $x$ is cubed. In $(5x)^3$ , both 5 and $x$ are cubed, giving $125x^3$ .

Explanation

This rule works because multiplication is commutative. An expression like $(2x)^3$ means $(2x)(2x)(2x)$ . You can regroup the factors as $(2 \cdot 2 \cdot 2)(x \cdot x \cdot x)$ , which is simply $2^3x^3$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Product and Quotient of Powers Properties

Property

Examples

Product Property: Multiply $10^4 \cdot 10^2$ .

$10^4 \cdot 10^2 = 10^{4+2} = 10^6$ .

Quotient Property: Divide $\frac{10^7}{10^3}$ .

$\frac{10^7}{10^3} = 10^{7-3} = 10^4$ .

Using Negative Exponents: Multiply $x^{-5} \cdot x^2$ .

$x^{-5} \cdot x^2 = x^{-5+2} = x^{-3}$ , which can be written as $\frac{1}{x^3}$ .

Explanation

Section 2

Power of a Power Property

Property

To raise a power to a power, keep the same base and multiply the exponents. In symbols,

(a^m)^n = a^{mn}

Examples

To simplify $(x^3)^5$ , you multiply the exponents: $x^{3 \cdot 5} = x^{15}$ .
To simplify $(4^2)^3$ , you keep the base and multiply the powers: $4^{2 \cdot 3} = 4^6$ .
Be careful to distinguish from products: $(a^5)(a^2) = a^{5+2} = a^7$ , but $(a^5)^2 = a^{5 \cdot 2} = a^{10}$ .

Explanation

Section 3

Power of a Product Property

Property

To raise a product to a power, raise each factor to the power. In symbols,

(ab)^n = a^nb^n

Examples

To simplify $(4xy)^2$ , apply the exponent to each factor inside: $4^2x^2y^2 = 16x^2y^2$ .
For $(-3a^2)^3$ , raise each factor to the third power: $(-3)^3(a^2)^3 = -27a^6$ .
Note the difference: in $5x^3$ , only $x$ is cubed. In $(5x)^3$ , both 5 and $x$ are cubed, giving $125x^3$ .

Explanation

Overview

Lesson overview

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

Overview

Section 1

Product and Quotient of Powers Properties

Property

Examples

Product Property: Multiply $10^4 \cdot 10^2$ .

$10^4 \cdot 10^2 = 10^{4+2} = 10^6$ .

Quotient Property: Divide $\frac{10^7}{10^3}$ .

$\frac{10^7}{10^3} = 10^{7-3} = 10^4$ .

Using Negative Exponents: Multiply $x^{-5} \cdot x^2$ .

$x^{-5} \cdot x^2 = x^{-5+2} = x^{-3}$ , which can be written as $\frac{1}{x^3}$ .

Explanation

Section 2

Power of a Power Property

Property

To raise a power to a power, keep the same base and multiply the exponents. In symbols,

(a^m)^n = a^{mn}

Examples

To simplify $(x^3)^5$ , you multiply the exponents: $x^{3 \cdot 5} = x^{15}$ .
To simplify $(4^2)^3$ , you keep the base and multiply the powers: $4^{2 \cdot 3} = 4^6$ .
Be careful to distinguish from products: $(a^5)(a^2) = a^{5+2} = a^7$ , but $(a^5)^2 = a^{5 \cdot 2} = a^{10}$ .

Explanation

Section 3

Power of a Product Property

Property

To raise a product to a power, raise each factor to the power. In symbols,

(ab)^n = a^nb^n

Examples

To simplify $(4xy)^2$ , apply the exponent to each factor inside: $4^2x^2y^2 = 16x^2y^2$ .
For $(-3a^2)^3$ , raise each factor to the third power: $(-3)^3(a^2)^3 = -27a^6$ .
Note the difference: in $5x^3$ , only $x$ is cubed. In $(5x)^3$ , both 5 and $x$ are cubed, giving $125x^3$ .