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

Lesson 1: Exponents

In this Grade 8 lesson from Big Ideas Math, Course 3, students learn how to write and evaluate expressions using integer exponents, including identifying the base and exponent of a power. Students practice converting repeated multiplication into exponential notation, working with negative bases such as (-3)^n, and distinguishing between expressions like (-2)^4 and -2^4. The lesson aligns with Common Core standard 8.EE.1 and builds foundational skills for the chapter's broader study of scientific notation.

Section 1

Exponential Notation

Property

For any expression $a^n$ , $a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.
The expression $a^n$ is read $a$ to the $n^{th}$ power. In $a^n$ , the $a$ is called the base and the $n$ is called the exponent.
$a^n = a \cdot a \cdot a \cdot ... \cdot a$ ( $n$ factors)
Special names are used for powers of 2 and 3:
$a^2$ is read as ' $a$ squared'
$a^3$ is read as ' $a$ cubed'

Examples

The expression $5 \cdot 5 \cdot 5 \cdot 5$ can be written in exponential notation as $5^4$ , where 5 is the base and 4 is the exponent.
To write $y^3$ in expanded form, you write out the base $y$ multiplied by itself 3 times: $y \cdot y \cdot y$ .
To simplify $2^5$ , you calculate $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ , which equals $32$ .

Explanation

Exponents are a shortcut for writing repeated multiplication. The small exponent number tells you how many times to multiply the larger base number by itself. This makes it much faster to write out long calculations involving the same factor.

Section 2

Products with Multiple Bases

Property

A product containing different repeated factors can be simplified by grouping like factors and writing each group as a power.

\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ factors}} \cdot \underbrace{b \cdot b \cdot \dots \cdot b}_{m \text{ factors}} = a^n b^m

Examples

Section 3

Exponential notation and order of operations

Property

Exponential Notation: $a^n$ means multiply $a$ by itself, $n$ times. The expression $a^n$ is read $a$ to the $n^{th}$ power. In $a^n$ , the $a$ is the base and the $n$ is the exponent.
Order of Operations:

Parentheses and other Grouping Symbols
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Examples

The expression $4^3$ is in exponential notation and means $4 \cdot 4 \cdot 4$ , which equals 64. Here, 4 is the base and 3 is the exponent.
To simplify $20 - 3 \cdot 5$ , we follow the order of operations by multiplying first: $20 - 15 = 5$ .
To simplify $2(5+1) + 3^2$ , first handle parentheses: $2(6) + 3^2$ . Next, exponents: $2(6) + 9$ . Then multiply: $12 + 9$ . Finally, add to get $21$ .

Explanation

Exponents are a shortcut for repeated multiplication. To ensure everyone gets the same answer for a problem, we follow the order of operations (PEMDAS/GEMDAS). It's the universal grammar for solving math expressions, ensuring consistent results.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Exponential Notation

Property

Examples

The expression $5 \cdot 5 \cdot 5 \cdot 5$ can be written in exponential notation as $5^4$ , where 5 is the base and 4 is the exponent.
To write $y^3$ in expanded form, you write out the base $y$ multiplied by itself 3 times: $y \cdot y \cdot y$ .
To simplify $2^5$ , you calculate $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ , which equals $32$ .

Explanation

Section 2

Products with Multiple Bases

Property

A product containing different repeated factors can be simplified by grouping like factors and writing each group as a power.

\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ factors}} \cdot \underbrace{b \cdot b \cdot \dots \cdot b}_{m \text{ factors}} = a^n b^m

Examples

Section 3

Exponential notation and order of operations

Property

Parentheses and other Grouping Symbols
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Examples

The expression $4^3$ is in exponential notation and means $4 \cdot 4 \cdot 4$ , which equals 64. Here, 4 is the base and 3 is the exponent.
To simplify $20 - 3 \cdot 5$ , we follow the order of operations by multiplying first: $20 - 15 = 5$ .
To simplify $2(5+1) + 3^2$ , first handle parentheses: $2(6) + 3^2$ . Next, exponents: $2(6) + 9$ . Then multiply: $12 + 9$ . Finally, add to get $21$ .

Explanation

Overview

Lesson overview

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

Overview

Section 1

Exponential Notation

Property

Examples

The expression $5 \cdot 5 \cdot 5 \cdot 5$ can be written in exponential notation as $5^4$ , where 5 is the base and 4 is the exponent.
To write $y^3$ in expanded form, you write out the base $y$ multiplied by itself 3 times: $y \cdot y \cdot y$ .
To simplify $2^5$ , you calculate $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ , which equals $32$ .

Explanation

Section 2

Products with Multiple Bases

Property

A product containing different repeated factors can be simplified by grouping like factors and writing each group as a power.

\underbrace{a \cdot a \cdot \dots \cdot a}_{n \text{ factors}} \cdot \underbrace{b \cdot b \cdot \dots \cdot b}_{m \text{ factors}} = a^n b^m

Examples

Section 3

Exponential notation and order of operations

Property

Parentheses and other Grouping Symbols
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Examples

The expression $4^3$ is in exponential notation and means $4 \cdot 4 \cdot 4$ , which equals 64. Here, 4 is the base and 3 is the exponent.
To simplify $20 - 3 \cdot 5$ , we follow the order of operations by multiplying first: $20 - 15 = 5$ .
To simplify $2(5+1) + 3^2$ , first handle parentheses: $2(6) + 3^2$ . Next, exponents: $2(6) + 9$ . Then multiply: $12 + 9$ . Finally, add to get $21$ .