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.

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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

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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

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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

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

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Questions & Step-by-Step Solutions

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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.

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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:

• 1
• 4
• 9
• 16
• 25

Final Answer

All square numbers have an odd number of factors.

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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

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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

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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.

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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.

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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.

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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.

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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

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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

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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

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

Solution

$\sqrt{441}=21$

Final Answer

21 m

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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

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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

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Q7. How many numbers lie between:

(i) 16² and 17²

Using formula:

2 × 16 = 32

Answer

32

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(ii) 99² and 100²

2 × 99 = 198

Answer

198

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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

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Page No. 13

Q9. Can a cube end with exactly two zeroes?

Solution

No. Cubes always contain zeros in multiples of 3.

Final Answer

No

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Page No. 14

Q10. Find the sum:

91 + 93 + 95 + … + 109

Solution

This pattern represents cube numbers.

Final Answer

$10^3=1000$

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Page No. 15

Q11. Find cube roots

(i)

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

(ii)

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

(iii)

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

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Page No. 16

Q12. Find the cube roots of 27000 and 10648

Solution

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

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

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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

7

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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

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Q15. Guess the cube roots

Answers

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

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

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

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

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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

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Common Mistakes

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

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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

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Practice MCQs

1. Which is a perfect square?

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

Answer

B. 64

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2. Cube root of 125 is:

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

Answer

B. 5

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3. Which number is not a perfect cube?

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

Answer

C. 81

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FAQ Section

Q1. What is a perfect square?

A number obtained by multiplying a number by itself.

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Q2. What is a perfect cube?

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

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Q3. Can a perfect square end with 7?

No.

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Q4. Can a cube end with 8?

Yes.

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Q5. What is the square root of 169?

$\sqrt{169}=13$

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