Square Numbers

A square number is the product of an integer multiplied by itself. Formally, for any integer \( n \), the square number is \( n^2 = n \times n \). Square numbers are also called perfect squares.

Formula Derivation

Given \( n \in \mathbb{Z} \), the square of \( n \) is \( n^2 \). For example, if \( n = 5 \), then \( 5^2 = 5 \times 5 = 25 \).

Worked Illustrations

Calculate the square of 7.

Step 1: Identify \( n = 7 \).

Step 2: Compute \( 7^2 = 7 \times 7 = 49 \).

Therefore, the square of 7 is 49.

Solved Examples

Example 1: Find the square of 12.

Step 1: \( n = 12 \).

Step 2: \( 12^2 = 12 \times 12 = 144 \).

Answer: 144

Example 2: Find the square of 15.

Step 1: \( n = 15 \).

Step 2: \( 15^2 = 15 \times 15 = 225 \).

Answer: 225

Practice Set

• Level 1 – Easy
• Find the square of 8.
• Find the square of 10.
• Find the square of 3.
• Level 2 – Moderate
• Find the square of 17.
• Find the square of 23.
• Find the square of 30.
• Level 3 – Challenging
• Find the square of 125.
• Find the square of 99.
• Find the square of 144.

Answer Key

• 8^2 = 64
• 10^2 = 100
• 3^2 = 9
• 17^2 = 289
• 23^2 = 529
• 30^2 = 900
• 125^2 = 15625
• 99^2 = 9801
• 144^2 = 20736

Quick Reference

Number (n)	Square (n^2)
1	1
2	4
3	9
4	16
5	25

Glossary

• Square Number: A number obtained by multiplying an integer by itself.
• Perfect Square: A square number that is the square of a natural number.
• Square Root: The inverse operation of squaring, denoted \( \sqrt{n} \).

Cube Numbers

A cube number is the product of an integer multiplied by itself three times. For any integer \( n \), the cube number is \( n^3 = n \times n \times n \). Cube numbers are also called perfect cubes.

Formula Derivation

Given \( n \in \mathbb{Z} \), the cube of \( n \) is \( n^3 \). For example, if \( n = 4 \), then \( 4^3 = 4 \times 4 \times 4 = 64 \).

Worked Illustrations

Calculate the cube of 5.

Step 1: Identify \( n = 5 \).

Step 2: Compute \( 5^3 = 5 \times 5 \times 5 = 125 \).

Therefore, the cube of 5 is 125.

Solved Examples

Example 1: Find the cube of 3.

Step 1: \( n = 3 \).

Step 2: \( 3^3 = 3 \times 3 \times 3 = 27 \).

Answer: 27

Example 2: Find the cube of 7.

Step 1: \( n = 7 \).

Step 2: \( 7^3 = 7 \times 7 \times 7 = 343 \).

Answer: 343

Practice Set

• Level 1 – Easy
• Find the cube of 2.
• Find the cube of 4.
• Find the cube of 1.
• Level 2 – Moderate
• Find the cube of 6.
• Find the cube of 9.
• Find the cube of 10.
• Level 3 – Challenging
• Find the cube of 15.
• Find the cube of 20.
• Find the cube of 25.

Answer Key

• 2^3 = 8
• 4^3 = 64
• 1^3 = 1
• 6^3 = 216
• 9^3 = 729
• 10^3 = 1000
• 15^3 = 3375
• 20^3 = 8000
• 25^3 = 15625

Quick Reference

Number (n)	Cube (n^3)
1	1
2	8
3	27
4	64
5	125

Glossary

• Cube Number: A number obtained by multiplying an integer by itself three times.
• Perfect Cube: A cube number that is the cube of a natural number.
• Cube Root: The inverse operation of cubing, denoted \( \sqrt[3]{n} \).