Short Introduction

This chapter explains important mathematical concepts like Perfect Squares, Square Roots, Perfect Cubes, and Cube Roots in a simple and logical way. Below are detailed step-by-step solutions for all important questions from the chapter “A Square and A Cube” in a portal-ready SEO format.

Quick Information Box

Particular	Details
Chapter Name	A Square and A Cube
Class	Grade 8
Subject	Mathematics
Main Topics	Perfect Squares, Cubes, Square Roots, Cube Roots
Difficulty Level	Easy to Moderate
Important For	School Exams, Olympiads, Scholarships
Learning Outcome	Understanding number patterns and roots

Concepts Used (Topics Covered)

Perfect Squares
Perfect Cubes
Square Roots
Cube Roots
Prime Factorisation
Properties of Squares
Properties of Cubes
Odd Number Patterns
Estimation of Roots
Number Patterns

Important Formulas

Square Formula

$n^2=n\times n$

Cube Formula

$n^3=n\times n\times n$ n3=n×n×n

Square Root

$\sqrt{n^2}=n$

Cube Root

$\sqrt[3]{n^3}=n$

Sum of Consecutive Odd Numbers

$1+3+5+\cdots +(2n-1)=n^2$

Questions & Step-by-Step Solutions

Page No. 2

Q1. Does every number have an even number of factors?

Solution

No. Perfect square numbers have an odd number of factors.

Example:

Factors of 4 → 1, 2, 4
Total factors = 3 (odd)

Final Answer

No, every number does not have an even number of factors.

Q2. Can you use this insight to find more numbers with an odd number of factors?

Solution

Yes. All perfect square numbers have an odd number of factors.

Examples:

Final Answer

All square numbers have an odd number of factors.

Page No. 3

Q3. Write the locker numbers that remain open.

Solution

Only lockers with perfect square numbers remain open because they are toggled an odd number of times.

The perfect squares up to 100 are:

$1,4,9,16,25,36,49,64,81,100$ 1,4,9,16,25,36,49,64,81,100

Final Answer

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Page No. 5

Q4. Which of the following numbers have the digit 6 in the units place?

Given:

38²
34²
46²
56²
74²
82²

Solution

A number ending in 4 or 6 gives a square ending in 6.

Therefore:

34² ✔
46² ✔
56² ✔
74² ✔

Final Answer

34², 46², 56², 74²

Q5. If a number contains 3 zeros at the end, how many zeros will its square have?

Solution

Example:

$1000^2=1000000$

The square has 6 zeros.

Final Answer

Six zeros.

Q6. What do you notice about zeros at the end of a number and its square?

Solution

The number of zeros at the end of a square is double the zeros at the end of the original number.

Final Answer

Squares always have an even number of zeros at the end.

Q7. What can you say about the parity of a number and its square?

Solution

Square of an even number is even.
Square of an odd number is odd.

Final Answer

Square of an even number is even and square of an odd number is odd.

Page No. 7

Q8. Find how many numbers lie between two consecutive perfect squares.

Solution

For consecutive perfect squares:

$n^2\text{ and }(n+1)^2$

Numbers between them:

$(n+1)^2-n^2-1=2n$

Final Answer

There are 2n numbers between consecutive perfect squares.

Figure It Out (Page 10)

Q1. Which of the following numbers are not perfect squares?

Numbers:

2032, 2048, 1027, 1089

Solution

1089 = 33²

The remaining numbers are not perfect squares.

Final Answer

2032, 2048, 1027

Q2. Which one among 64², 108², 292², 36² has last digit 4?

Solution

Numbers ending in 2 or 8 have squares ending in 4.

Final Answer

108² and 292²

Q3. Given 125² = 15625, find 126²

Solution

Using identity:

$(a+1)^2=a^2+2a+1$

126² = 125² + 2×125 +1

= 15625 + 251

= 15876

Final Answer

15876

Q4. Find the side of a square whose area is 441 m²

Solution

$\sqrt{441}=21$

Final Answer

21 m

Q5. Find the smallest square number divisible by 4, 9 and 10

Solution

LCM of 4, 9 and 10:

= 180

Prime factors:

180 = 2² × 3² × 5

To make it a perfect square, multiply by another 5.

180 × 5 = 900

Final Answer

900

Q6. Find the smallest number by which 9408 must be multiplied to make a perfect square

Solution

Prime factorisation:

9408 = 2⁶ × 3 × 7²

3 has no pair.

Multiply by 3.

9408 × 3 = 28224

Square root:

$\sqrt{28224}=168$

Final Answer

Multiplier = 3
Square Root = 168

Q7. How many numbers lie between:

(i) 16² and 17²

Using formula:

2 × 16 = 32

Answer

(ii) 99² and 100²

2 × 99 = 198

Answer

198

Q8. Fill in the missing numbers

Solution

$4^2+5^2+20^2=21^2$

$9^2+10^2+90^2=91^2$

Final Answers

21, 90, 91

Page No. 13

Q9. Can a cube end with exactly two zeroes?

Solution

No. Cubes always contain zeros in multiples of 3.

Final Answer

Page No. 14

Q10. Find the sum:

91 + 93 + 95 + … + 109

Solution

This pattern represents cube numbers.

Final Answer

$10^3=1000$

Page No. 15

Q11. Find cube roots

(i)

$\sqrt[3]{64}=4$

(ii)

$\sqrt[3]{512}=8$

(iii)

$\sqrt[3]{729}=9$

Page No. 16

Q12. Find the cube roots of 27000 and 10648

Solution

$\sqrt[3]{27000}=30$

$\sqrt[3]{10648}=22$

Q13. What number will you multiply by 1323 to make it a cube number?

Solution

1323 = 3³ × 7²

One more 7 is required.

Final Answer

Q14. State True or False

Statement	Answer
Cube of any odd number is even	False
No perfect cube ends with 8	False
Cube of a 2-digit number may be a 3-digit number	False
Cube of a 2-digit number may have seven or more digits	False
Cube numbers have odd number of factors	False

Q15. Guess the cube roots

Answers

$\sqrt[3]{1331}=11$

$\sqrt[3]{4913}=17$

$\sqrt[3]{12167}=23$

$\sqrt[3]{32768}=32$

Page No. 17

Q16. Which of the following is greatest?

Solution

Cube differences increase rapidly.

Therefore:

$67^3-66^3$

is the greatest.

Final Answer

67³ − 66³

Common Mistakes

Confusing square numbers with cube numbers
Incorrect prime factorisation
Forgetting pairing and triplet grouping
Wrong estimation of roots
Ignoring units digit rules

Exam Tips

Memorise squares up to 30
Memorise cubes up to 20
Practice prime factorisation daily
Learn units digit shortcuts
Revise odd number patterns carefully

Practice MCQs

1. Which is a perfect square?

A. 48
B. 64
C. 72
D. 98

Answer

B. 64

2. Cube root of 125 is:

A. 4
B. 5
C. 6
D. 7

Answer

B. 5

3. Which number is not a perfect cube?

A. 27
B. 64
C. 81
D. 125

Answer

C. 81

FAQ Section

Q1. What is a perfect square?

A number obtained by multiplying a number by itself.

Q2. What is a perfect cube?

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

Q3. Can a perfect square end with 7?

No.

Q4. Can a cube end with 8?

Yes.

Q5. What is the square root of 169?

$\sqrt{169}=13$

📘 Get more NCERT Solutions, Mock Tests, Practice Questions, and Exam Preparation Resources at:

www.mymockmate.com

🚀 Improve your preparation with All India Mock Tests and Instant Results.

NCERT Class 8 A Square and A Cube Solutions Guide

Short Introduction

Quick Information Box

Concepts Used (Topics Covered)

Important Formulas

Square Formula

Cube Formula

Square Root

Cube Root

Sum of Consecutive Odd Numbers

Questions & Step-by-Step Solutions

Page No. 2

Q1. Does every number have an even number of factors?

Solution

Final Answer

Q2. Can you use this insight to find more numbers with an odd number of factors?

Solution

Final Answer

Page No. 3

Q3. Write the locker numbers that remain open.

Solution

Final Answer

Page No. 5

Q4. Which of the following numbers have the digit 6 in the units place?

Solution

Final Answer

Q5. If a number contains 3 zeros at the end, how many zeros will its square have?

Solution

Final Answer

Q6. What do you notice about zeros at the end of a number and its square?

Solution

Final Answer

Q7. What can you say about the parity of a number and its square?

Solution

Final Answer

Page No. 7

Q8. Find how many numbers lie between two consecutive perfect squares.

Solution

Final Answer

Figure It Out (Page 10)

Q1. Which of the following numbers are not perfect squares?

Solution

Final Answer

Q2. Which one among 64², 108², 292², 36² has last digit 4?

Solution

Final Answer

Q3. Given 125² = 15625, find 126²

Solution

Final Answer

Q4. Find the side of a square whose area is 441 m²

Solution

Final Answer

Q5. Find the smallest square number divisible by 4, 9 and 10

Solution

Final Answer

Q6. Find the smallest number by which 9408 must be multiplied to make a perfect square

Solution

Final Answer

Q7. How many numbers lie between:

(i) 16² and 17²

Answer

(ii) 99² and 100²

Answer

Q8. Fill in the missing numbers

Solution

Final Answers

Page No. 13

Q9. Can a cube end with exactly two zeroes?

Solution

Final Answer

Page No. 14

Q10. Find the sum:

Solution

Final Answer

Page No. 15

Q11. Find cube roots

(i)

(ii)

(iii)

Page No. 16