Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 4: Real Numbers and the Pythagorean Theorem

Lesson 1: Finding Square Roots

In this Grade 7 lesson from Big Ideas Math, Course 2 Accelerated (Chapter 4), students learn how to find square roots of perfect squares using the radical sign and apply them to solve for unknown dimensions. Students practice evaluating expressions involving square roots and using square roots to solve equations of the form x² = p, including cases with fractions and decimals. Real-world applications include finding the side length of a square and the radius of a circle when given the area.

Section 1

Square Root of a Number

Property

Square of a Number
If $n^2 = m$ , then $m$ is the square of $n$ .

Square Root of a Number
If $n^2 = m$ , then $n$ is a square root of $m$ .

Square Root Notation
$\sqrt{m}$ is read as “the square root of $m$ .”
If $m = n^2$ , then $\sqrt{m} = n$ , for $n \ge 0$ .
The square root of $m$ , $\sqrt{m}$ , is the positive number whose square is $m$ . This is also called the principal square root.
To find the negative square root of a number, we place a negative in front of the radical sign.
When using the order of operations, we treat the radical as a grouping symbol.

Section 2

Vocabulary: Radical, Radicand, and Principal Root

Property

The symbol $\sqrt{\ }$ is called a radical sign, and the number inside is called the radicand.
The positive square root of a number is called the principal square root.

Examples

The principal square root of 81 is written as $\sqrt{81}$ , which equals 9.
To express the negative square root of 36, we write $-\sqrt{36}$ , which equals $-6$ .
The expression $\pm\sqrt{100}$ represents both square roots of 100, which means 10 or $-10$ .

Explanation

The radical symbol $\sqrt{\ }$ is a specific instruction to find only the positive square root, known as the principal root.

Section 3

Square Roots and Perfect Squares

Property

The symbol $\sqrt{A}$ (square root) indicates a number $a$ whose square is $A$ : $a^2 = A$ . The square root $\sqrt{A}$ is only defined for non-negative numbers $A$ . A positive integer whose square root is a positive integer is called a perfect square.

Examples

Since $8^2 = 64$ , the square root of 64 is 8. We write this as $\sqrt{64} = 8$ . The number 64 is a perfect square.
The number 50 is not a perfect square. Its square root, $\sqrt{50}$ , is a number that, when multiplied by itself, equals 50.
To find the number whose square is 121, we are looking for $\sqrt{121}$ . Since $11 \times 11 = 121$ , the answer is $11$ .

Explanation

A square root is the opposite of squaring a number. If you know the area of a square, the square root tells you the side length. Perfect squares are special because their square roots are nice, neat whole numbers!

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Square Root of a Number

Property

Square of a Number
If $n^2 = m$ , then $m$ is the square of $n$ .

Square Root of a Number
If $n^2 = m$ , then $n$ is a square root of $m$ .

Section 2

Vocabulary: Radical, Radicand, and Principal Root

Property

The symbol $\sqrt{\ }$ is called a radical sign, and the number inside is called the radicand.
The positive square root of a number is called the principal square root.

Examples

The principal square root of 81 is written as $\sqrt{81}$ , which equals 9.
To express the negative square root of 36, we write $-\sqrt{36}$ , which equals $-6$ .
The expression $\pm\sqrt{100}$ represents both square roots of 100, which means 10 or $-10$ .

Explanation

The radical symbol $\sqrt{\ }$ is a specific instruction to find only the positive square root, known as the principal root.

Section 3

Square Roots and Perfect Squares

Property

Examples

Since $8^2 = 64$ , the square root of 64 is 8. We write this as $\sqrt{64} = 8$ . The number 64 is a perfect square.
The number 50 is not a perfect square. Its square root, $\sqrt{50}$ , is a number that, when multiplied by itself, equals 50.
To find the number whose square is 121, we are looking for $\sqrt{121}$ . Since $11 \times 11 = 121$ , the answer is $11$ .

Explanation

Overview

Lesson overview

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

Overview

Section 1

Square Root of a Number

Property

Square of a Number
If $n^2 = m$ , then $m$ is the square of $n$ .

Square Root of a Number
If $n^2 = m$ , then $n$ is a square root of $m$ .

Section 2

Vocabulary: Radical, Radicand, and Principal Root

Property

The symbol $\sqrt{\ }$ is called a radical sign, and the number inside is called the radicand.
The positive square root of a number is called the principal square root.

Examples

The principal square root of 81 is written as $\sqrt{81}$ , which equals 9.
To express the negative square root of 36, we write $-\sqrt{36}$ , which equals $-6$ .
The expression $\pm\sqrt{100}$ represents both square roots of 100, which means 10 or $-10$ .

Explanation

The radical symbol $\sqrt{\ }$ is a specific instruction to find only the positive square root, known as the principal root.

Section 3

Square Roots and Perfect Squares

Property

Examples

Since $8^2 = 64$ , the square root of 64 is 8. We write this as $\sqrt{64} = 8$ . The number 64 is a perfect square.
The number 50 is not a perfect square. Its square root, $\sqrt{50}$ , is a number that, when multiplied by itself, equals 50.
To find the number whose square is 121, we are looking for $\sqrt{121}$ . Since $11 \times 11 = 121$ , the answer is $11$ .