Squares and Square Roots

For Elementary Students

What Is Squaring a Number?

Squaring means multiplying a number by itself.

Think about it like this: If you have a square with sides of 3, the area is 3 × 3 = 9. That's squaring!

How to Write a Square

We write "3 squared" as 3²

The little 2 up high means "multiply this number by itself."

Examples:

• 2² = 2 × 2 = 4 (read as "2 squared equals 4")
• 4² = 4 × 4 = 16 ("4 squared equals 16")
• 5² = 5 × 5 = 25 ("5 squared equals 25")

Why Is It Called "Squared"?

Imagine making a square!

Example: A square with sides of 3

▢▢▢
▢▢▢  →  3 × 3 = 9 squares total
▢▢▢

The area is 9, so 3² = 9!

Perfect Squares You Should Know

These are squares of whole numbers. Try to memorize them!

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• 6² = 36
• 7² = 49
• 8² = 64
• 9² = 81
• 10² = 100

What Is a Square Root?

A square root is the opposite of squaring. It asks: "What number was squared to get this?"

Symbol: √ (called a "radical sign")

Example: √9 = 3

Why? Because 3 × 3 = 9

Think of it like this: Squaring makes a number bigger. Square root undoes it!

More Square Root Examples

• √4 = 2 (because 2 × 2 = 4)
• √16 = 4 (because 4 × 4 = 16)
• √25 = 5 (because 5 × 5 = 25)
• √100 = 10 (because 10 × 10 = 100)

Squaring and Square Roots Are Opposites!

Going one way: 5² = 25 (squaring)

Going back: √25 = 5 (square root)

They undo each other!

For Junior High Students

Understanding Squaring

Squaring a number means raising it to the power of 2.

General form: n² = n × n

Examples:

• 3² = 3 × 3 = 9
• 7² = 7 × 7 = 49
• 10² = 10 × 10 = 100
• 12² = 12 × 12 = 144

Perfect Squares

A perfect square is any number that equals a whole number squared.

List of perfect squares to memorize:

Tip: Knowing these by heart makes many math problems much faster!

Understanding Square Roots

The square root operation is the inverse of squaring.

Definition: √a = b means b² = a

Symbol: √ (radical sign)

Examples:

• √9 = 3 because 3² = 9
• √25 = 5 because 5² = 25
• √144 = 12 because 12² = 144

Finding Square Roots of Perfect Squares

If you know your perfect squares, finding square roots is easy!

Example: √49 = ?

Think: "What number squared equals 49?"

• You know 7² = 49
• Answer: √49 = 7

Estimating Square Roots of Non-Perfect Squares

Not all numbers are perfect squares! For these, estimate.

Example: Estimate √20

Step 1: Find perfect squares on either side

• √16 = 4
• √25 = 5

Step 2: 20 is between 16 and 25, so √20 is between 4 and 5

Step 3: 20 is closer to 16, so √20 is closer to 4 (about 4.5)

Calculator says: √20 ≈ 4.47

Squaring Negative Numbers

When you square a negative number, you get a positive!

Why? Negative × negative = positive

Examples:

• (-3)² = (-3) × (-3) = 9
• (-5)² = (-5) × (-5) = 25
• (-10)² = (-10) × (-10) = 100

Important: All perfect squares are positive (or zero)!

Be Careful with Negative Signs!

With parentheses: (-4)² = (-4) × (-4) = 16

Without parentheses: -4² = -(4²) = -(16) = -16

The parentheses matter!

Square Roots of Negative Numbers

You cannot take the square root of a negative number (in regular math)!

❌ √(-9) has no real answer

Why? No real number times itself gives a negative!

• 3 × 3 = 9 (positive)
• (-3) × (-3) = 9 (positive!)

We will learn about imaginary numbers later.

Using Square Roots in Geometry

Finding side length from area:

Example: A square has area 64 square feet. What is the side length?

• Area = side²
• 64 = side²
• side = √64 = 8 feet

Pythagorean theorem uses squares and square roots!

In a right triangle: a² + b² = c²

Example: Legs are 3 and 4. Find hypotenuse.

• 3² + 4² = c²
• 9 + 16 = c²
• 25 = c²
• c = √25 = 5

Properties of Squares and Square Roots

Property 1: √(a²) = a (square root undoes square)

Property 2: (√a)² = a (square undoes square root)

Property 3: √(a × b) = √a × √b

Example: √36 = √(4 × 9) = √4 × √9 = 2 × 3 = 6

Real-Life Uses

Area: "Square garden with area 49 m². Side length?"

• √49 = 7 meters

Physics: Many formulas use squares (distance, energy, etc.)

Construction: Checking if corners are square (3-4-5 triangle)

Statistics: Standard deviation uses squares and square roots

Practice