March 11, 2026

Class 8 Maths Chapter 1: A Square and A Cube Full Solutions & FAQs

Class 8 Maths – Chapter 1: A Square and A Cube (Ganita Prakash Part 1)

Introduction

This chapter explores the beautiful patterns behind square numbers, cube numbers, and their roots. Students learn how to identify perfect squares and cubes using factorisation, patterns, and visual reasoning. The chapter also includes puzzles, difference patterns, and real‑life applications.

Basic Knowledge Required

Multiplication tables
Factors and multiples
Prime factorisation
Basic geometry (square, cube)
Odd and even numbers

Important Definitions

Square Number

A number obtained by multiplying a number by itself: n² = n × n

Perfect Square

A square number formed from a natural number. Examples: 1, 4, 9, 16, 25…

Cube Number

A number obtained by multiplying a number by itself three times: n³ = n × n × n

Perfect Cube

A cube number formed from a natural number. Examples: 1, 8, 27, 64…

Square Root

√n = x if x² = n

Cube Root

∛n = x if x³ = n

Formulas Used

Concept	Formula
Square of a number	n² = n × n
Cube of a number	n³ = n × n × n
nth odd number	2n − 1
Square root by factorisation	Pair prime factors
Cube root by factorisation	Group prime factors in triplets
Identity	(n+1)² = n² + 2n + 1

Solved Examples (Step‑by‑Step)

Example 1: Find the square of 36.

(30 + 6)² = 900 + 360 + 36 = 1296

Example 2: Check if 156 is a perfect square.

156 = 2² × 3 × 13 → unpaired factors → Not a perfect square

Example 3: Find the cube of 7.

7³ = 343

Example 4: Check if 3375 is a perfect cube.

3375 = 3³ × 5³ → triplets possible → Perfect cube

∛3375 = 15

Figure It Out – Complete Solutions
SECTION A — PERFECT SQUARES

1. Which numbers are NOT perfect squares?

2032, 2048, 1027 are not perfect squares. 1089 = 33² → perfect square.

2. Which number has last digit 4?

108² and 292²

3. Given 125² = 15625, find 126².

126² = 15625 + 250 + 1 = 15876

4. Find the side of a square whose area is 441 m².

√441 = 21 m

5. Smallest square divisible by 4, 9, 10.

LCM = 180 → multiply by 5 → 900

6. Smallest number to multiply 9408 to make it a perfect square.

9408 = 2⁶ × 3 × 7 → multiply by 21 → perfect square = 197568

√197568 = 168

7. Numbers between squares

Between 16² and 17² → 32

Between 99² and 100² → 198

8. Fill in the pattern

4² + 5² + 20² = 21²

9² + 10² + 27² = 28²

9. Tiny squares

SECTION B — PERFECT CUBES

1. Cube roots of 27000 and 10648

∛27000 = 30

∛10648 = 22

2. Number to multiply 1323 to make it a cube

1323 = 3³ × 7² → multiply by 7 → cube number

3. True or False

Cube of odd number is even → False
No cube ends with 8 → False
Cube of 2‑digit number may be 3‑digit → True
Cube of 2‑digit number may have 7+ digits → True
Cube numbers have odd number of factors → False

4. Guess cube roots

1331 → 11

4913 → 17

12167 → 23

32768 → 32

5. Greatest value

67³ − 66³ is the greatest.

FAQs (Clickable Toggles)

What is a perfect square?

A number obtained by multiplying a number by itself. Example: 49 = 7².

How do we identify a perfect square?

Check if prime factors can be paired completely.

Which digits cannot appear at the end of a square?

2, 3, 7, 8.

What is the nth odd number?

2n − 1.

How to find square root by subtraction?

Subtract consecutive odd numbers until you reach 0.

What is a perfect cube?

A number obtained by multiplying a number by itself three times.

How to check if a number is a perfect cube?

Prime factors must form triplets.

What is the cube root of 3375?

15.

Can a cube end with 00?

No. A cube needs three 2s and three 5s.

What is the smallest square divisible by 4, 9, 10?

900.

How many numbers lie between n² and (n+1)²?

2n.

What is the cube of 12?

1728.

What is the square root of 1936?

44.

Is 500 a perfect cube?

No.

What is the cube root of 10648?

22.

Conclusion

This chapter builds a strong foundation for understanding squares, cubes, and their roots using patterns, factorisation, and visual reasoning. These concepts are essential for algebra and higher mathematics.