Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 2: Exponents

Lesson 1: Squares

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra (AMC 8), students learn how to square numbers, understand perfect squares, and apply the correct order of operations when evaluating expressions with exponents. The lesson covers key properties including the square of a product, quotient, and negation, with special attention to the difference between expressions like (-2)² and -2². Students practice these concepts through problems involving variable substitution and multi-step simplification.

Section 1

Square of a number

Property

If $n^2 = m$ , then $m$ is the square of $n$ . A perfect square is the square of a whole number. Squaring a positive or negative number results in a positive number.

Examples

The square of 8 is 64, because $8^2 = 8 \times 8 = 64$ .
The number 144 is a perfect square, as it is the square of a whole number: $12^2 = 144$ .
The square of $-5$ is 25, since $(-5)^2 = (-5)(-5) = 25$ .

Explanation

Squaring a number means multiplying it by itself. This concept is named after a geometric square, where the area is the side length multiplied by itself. Perfect squares are the result of squaring whole numbers, like 9 from $3^2$ .

Section 2

Order of Operations

Property

Step 1. Parentheses and Other Grouping Symbols
Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
Step 2. Exponents
Simplify all expressions with exponents.
Step 3. Multiplication and Division
Perform all multiplication and division in order from left to right. These operations have equal priority.
Step 4. Addition and Subtraction
Perform all addition and subtraction in order from left to right. These operations have equal priority.

Examples

To simplify $30 - 5 \cdot 4$ , we perform multiplication first: $30 - 20 = 10$ .
In the expression $(3+2)^2 \div 5$ , we start with parentheses $(5)^2 \div 5$ , then the exponent $25 \div 5$ , and finally division to get $5$ .
For $4 + 2[10 - 3(2)]$ , we work inside the innermost parentheses first: $4 + 2[10 - 6]$ , then inside the brackets $4 + 2[4]$ , then multiply $4 + 8$ , and finally add to get $12$ .

Explanation

The order of operations (PEMDAS) is a set of rules everyone follows to solve math problems. This ensures that every expression has only one correct answer, preventing confusion and making sure our calculations are consistent.

Section 3

Squares of Negative Numbers

Property

To show that a negative number is squared, we enclose the negative number in parentheses. For example, to indicate the square of $-5$ , we write

(-5)^2 = (-5)(-5) = 25

If the negative number is not enclosed in parentheses, the exponent applies only to the positive number.

-5^2 = -(5 \cdot 5) = -25

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Square of a number

Property

If $n^2 = m$ , then $m$ is the square of $n$ . A perfect square is the square of a whole number. Squaring a positive or negative number results in a positive number.

Examples

The square of 8 is 64, because $8^2 = 8 \times 8 = 64$ .
The number 144 is a perfect square, as it is the square of a whole number: $12^2 = 144$ .
The square of $-5$ is 25, since $(-5)^2 = (-5)(-5) = 25$ .

Explanation

Section 2

Order of Operations

Property

Examples

To simplify $30 - 5 \cdot 4$ , we perform multiplication first: $30 - 20 = 10$ .
In the expression $(3+2)^2 \div 5$ , we start with parentheses $(5)^2 \div 5$ , then the exponent $25 \div 5$ , and finally division to get $5$ .
For $4 + 2[10 - 3(2)]$ , we work inside the innermost parentheses first: $4 + 2[10 - 6]$ , then inside the brackets $4 + 2[4]$ , then multiply $4 + 8$ , and finally add to get $12$ .

Explanation

Section 3

Squares of Negative Numbers

Property

To show that a negative number is squared, we enclose the negative number in parentheses. For example, to indicate the square of $-5$ , we write

(-5)^2 = (-5)(-5) = 25

If the negative number is not enclosed in parentheses, the exponent applies only to the positive number.

-5^2 = -(5 \cdot 5) = -25

Examples

Overview

Lesson overview

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

Overview

Section 1

Square of a number

Property

If $n^2 = m$ , then $m$ is the square of $n$ . A perfect square is the square of a whole number. Squaring a positive or negative number results in a positive number.

Examples

The square of 8 is 64, because $8^2 = 8 \times 8 = 64$ .
The number 144 is a perfect square, as it is the square of a whole number: $12^2 = 144$ .
The square of $-5$ is 25, since $(-5)^2 = (-5)(-5) = 25$ .

Explanation

Section 2

Order of Operations

Property

Examples

To simplify $30 - 5 \cdot 4$ , we perform multiplication first: $30 - 20 = 10$ .
In the expression $(3+2)^2 \div 5$ , we start with parentheses $(5)^2 \div 5$ , then the exponent $25 \div 5$ , and finally division to get $5$ .
For $4 + 2[10 - 3(2)]$ , we work inside the innermost parentheses first: $4 + 2[10 - 6]$ , then inside the brackets $4 + 2[4]$ , then multiply $4 + 8$ , and finally add to get $12$ .

Explanation

Section 3

Squares of Negative Numbers

Property

To show that a negative number is squared, we enclose the negative number in parentheses. For example, to indicate the square of $-5$ , we write

(-5)^2 = (-5)(-5) = 25

If the negative number is not enclosed in parentheses, the exponent applies only to the positive number.

-5^2 = -(5 \cdot 5) = -25