A Square and A Cube explains numbers formed by multiplying a number by itself two times or three times. It teaches perfect squares, perfect cubes, square roots, cube roots, patterns and factor-based reasoning.
Important Questions Class 8 Maths Chapter 1 help students revise the 100-locker puzzle, square numbers, cube numbers, roots, prime factorisation and number patterns. This chapter builds the base for exponents, algebra, mensuration and higher-level number theory.
Class 8 Maths Chapter 1 begins with Queen Ratnamanjuri’s locker puzzle. The puzzle looks like a game, but it teaches a powerful idea: factors decide how often a number gets toggled.
Students then learn why only square-numbered lockers remain open, why perfect squares have special last digits, and how square roots connect to the side of a square. The chapter also introduces cubes through unit cubes, cube roots, consecutive odd-number patterns and the Hardy-Ramanujan number 1729.
Key Takeaways from Class 8 Maths Chapter 1
| Topic |
What Students Must Know |
| Chapter Name |
A Square and A Cube |
| Main Concepts |
Perfect squares, square roots, perfect cubes, cube roots |
| Opening Idea |
100-locker puzzle and factors |
| Square Number |
A number written as (n \times n) |
| Cube Number |
A number written as (n \times n \times n) |
| Important Method |
Prime factorisation for roots |
| Exam Focus |
Patterns, roots, factor pairs, estimation and reasoning |
Important Questions Class 8 Maths Chapter 1 with Answers
These questions cover the main ideas of the chapter. Focus on the reason behind each answer, not only the final number.
Important Questions Class 8 Maths Chapter 1: Basic Concepts
Q1. What is a square number?
A square number is a number obtained by multiplying a number by itself.
For example:
5 × 5 = 25
So, 25 is a square number.
Q2. What is a perfect square?
A perfect square is the square of a natural number.
Examples are:
1, 4, 9, 16, 25, 36
These are squares of 1, 2, 3, 4, 5 and 6.
Q3. What is a cube number?
A cube number is a number obtained by multiplying a number by itself three times.
For example:
4 × 4 × 4 = 64
So, 64 is a cube number.
Q4. What is a perfect cube?
A perfect cube is the cube of a natural number.
Examples are:
1, 8, 27, 64, 125
These are cubes of 1, 2, 3, 4 and 5.
Q5. Why do square-numbered lockers remain open in the 100-locker puzzle?
Square-numbered lockers remain open because they have an odd number of factors.
Most factors occur in pairs. A square number has one repeated factor pair, such as:
6 × 6 = 36
This repeated factor gives the number an odd number of factors.
Class 8 Maths Chapter 1 Question Answer on Square Numbers
Square numbers appear in area, number patterns and factor questions. Students should learn both the definition and the pattern.
Square Numbers Class 8 Questions
Q1. Write the first ten square numbers.
The first ten square numbers are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
They are:
1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2
Q2. Why are 1, 4, 9 and 16 called square numbers?
They are called square numbers because they can form square arrangements.
For example, 16 unit squares can form a square of side 4 units.
4 × 4 = 16
Q3. Which units digits are possible for perfect squares?
Perfect squares can end in:
0, 1, 4, 5, 6, 9
They cannot end in:
2, 3, 7, 8
Q4. Can a number ending in 6 always be a square?
No, a number ending in 6 need not always be a square.
For example, 16 and 36 are squares. But 26 is not a square.
Q5. Why can a perfect square not have an odd number of zeros at the end?
A perfect square has an even number of zeros at the end.
For example:
10^2 = 100
100^2 = 10000
One zero in the number becomes two zeros in the square.
Perfect Squares Class 8 Important Questions
Perfect squares can be tested through units digits, odd-number sums and prime factorisation. These questions help students avoid guesswork.
Q1. Is 2032 a perfect square?
No, 2032 is not a perfect square.
It ends in 2. A perfect square cannot end in 2.
Q2. Is 2048 a perfect square?
No, 2048 is not a perfect square.
It ends in 8. A perfect square cannot end in 8.
Q3. Is 1089 a perfect square?
Yes, 1089 is a perfect square.
33 × 33 = 1089
So:
1089 = 33^2
Q4. Given (125^2 = 15625), find (126^2).
Use the next-square pattern.
The difference between consecutive squares is the sum of the two numbers.
126^2 - 125^2 = 126 + 125
= 251
So:
126^2 = 15625 + 251
= 15876
Q5. How many numbers lie between (16^2) and (17^2)?
Between (n^2) and ((n+1)^2), there are (2n) natural numbers.
For (n = 16):
2 × 16 = 32
So, 32 numbers lie between (16^2) and (17^2).
Square Root Class 8 Questions with Answers
Square root means finding the side of a square when the area is known. It reverses the square operation.
Q1. What is square root?
Square root is the number which gives the original number when multiplied by itself.
For example:
7 × 7 = 49
So, the square root of 49 is 7.
Q2. Find the square root of 64.
8 × 8 = 64
So:
√(64) = 8
Q3. Find the side of a square whose area is 441 m².
The side of the square is:
√(441)
21 × 21 = 441
So, the side is 21 m.
Q4. Is 324 a perfect square? Find its square root.
Prime factorisation:
324 = 2 × 2 × 3 × 3 × 3 × 3
Pair the factors:
324 = (2 × 3 × 3)^2
= 18^2
So:
√(324) = 18
Q5. Is 156 a perfect square?
No, 156 is not a perfect square.
Prime factorisation:
156 = 2 × 2 × 3 × 13
The factors cannot form two identical groups.
Square Root Estimation Class 8 Extra Questions
Not every number is a perfect square. Estimation helps students find the closest square root value.
Q1. Estimate (√(250)).
We know:
15^2 = 225
16^2 = 256
So:
15 < √(250) < 16
Since 250 is closer to 256, (√(250)) is close to 16.
Q2. Find the largest square handkerchief from cloth of area 125 cm².
We need the largest integer side length.
11^2 = 121
12^2 = 144
Since 144 is greater than 125, the largest side is 11 cm.
Q3. Estimate (√(1936)).
1936 lies between:
40^2 = 1600
and
50^2 = 2500
Since it ends in 6, the square root may end in 4 or 6.
44^2 = 1936
So:
√(1936) = 44
Q4. Find the smallest square number divisible by 4, 9 and 10.
Prime factorisation:
4 = 2^2
9 = 3^2
10 = 2 × 5
LCM:
2^2 × 3^2 × 5 = 180
To make it a perfect square, multiply by 5.
180 × 5 = 900
So, the smallest square number is 900.
Q5. Find the smallest number by which 9408 must be multiplied to make a perfect square.
Prime factorisation:
9408 = 2^6 × 3 × 7^2
The factor 3 is unpaired.
So, multiply by 3.
9408 × 3 = 28224
√(28224) = 168
100-Locker Puzzle Questions from A Square and A Cube Class 8
The 100-locker puzzle is one of the most important reasoning parts of the chapter. It links factors, square numbers and odd-even toggling.
Q1. Which lockers remain open in the 100-locker puzzle?
The open lockers are square-numbered lockers.
They are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Q2. Why do only these lockers remain open?
A locker changes position once for every factor of its number.
Most factors come in pairs, so the locker gets toggled an even number of times. Square numbers have one repeated factor, so they have an odd number of factors.
Q3. What are the first five lockers touched exactly twice?
The lockers touched exactly twice are prime-numbered lockers.
The first five are:
2, 3, 5, 7, 11
A prime number has exactly two factors: 1 and itself.
Q4. How many lockers remain open from 1 to 100?
There are 10 square numbers from 1 to 100.
So, 10 lockers remain open.
Q5. Which locker remains open between 80 and 90?
The square number between 80 and 90 is 81.
9^2 = 81
So, locker 81 remains open.
Cube Numbers Class 8 Important Questions
Cube numbers connect arithmetic with 3D geometry. A cube of side (n) has (n^3) unit cubes.
Q1. Write the first ten cube numbers.
The first ten cube numbers are:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
They are cubes of 1 to 10.
Q2. Why is 64 a cube number?
64 is a cube number because:
4 × 4 × 4 = 64
A cube of side 4 units contains 64 unit cubes.
Q3. Is 9 a perfect cube?
No, 9 is not a perfect cube.
2^3 = 8
3^3 = 27
There is no natural number whose cube is 9.
Q4. Can a perfect cube end with 8?
Yes, a perfect cube can end with 8.
For example:
2^3 = 8
12^3 = 1728
Q5. Can a cube end with exactly two zeros?
No, a perfect cube cannot end with exactly two zeros.
If a number has one zero at the end, its cube has three zeros at the end. Zeros in cube numbers occur in multiples of 3.
Perfect Cubes Class 8 Questions with Answers
Perfect cubes can be checked through prime factorisation. Each prime factor must form a triplet.
Q1. Is 3375 a perfect cube?
Yes, 3375 is a perfect cube.
Prime factorisation:
3375 = 3 × 3 × 3 × 5 × 5 × 5
= 3^3 × 5^3
= (3 × 5)^3
= 15^3
Q2. Is 500 a perfect cube?
No, 500 is not a perfect cube.
Prime factorisation:
500 = 2 × 2 × 5 × 5 × 5
The factors cannot form triplets.
Q3. What number should multiply 1323 to make it a cube number?
Prime factorisation:
1323 = 3^3 × 7^2
To form triplets, we need one more 7.
So, multiply by 7.
1323 × 7 = 9261
9261 = 21^3
Q4. Is the cube of any odd number odd?
Yes, the cube of any odd number is odd.
Odd × odd × odd gives an odd product.
Q5. Can the cube of a 2-digit number have seven or more digits?
No, the cube of a 2-digit number cannot have seven or more digits.
The largest 2-digit number is 99.
99^3 = 970299
It has 6 digits.
Cube Root Class 8 Extra Questions
Cube root reverses the cube operation. Prime factorisation helps find cube roots of large numbers.
Q1. What is cube root?
Cube root is the number which gives the original number when multiplied by itself three times.
For example:
2 × 2 × 2 = 8
So:
sqrt[3]{8} = 2
Q2. Find (sqrt[3]{64}).
4 × 4 × 4 = 64
So:
sqrt[3]{64} = 4
Q3. Find (sqrt[3]{512}).
512 = 2^9
= (2^3)^3
= 8^3
So:
sqrt[3]{512} = 8
Q4. Find (sqrt[3]{729}).
729 = 9^3
So:
sqrt[3]{729} = 9
Q5. Find (sqrt[3]{27000}).
Prime factorisation:
27000 = 27 × 1000
= 3^3 × 10^3
= 30^3
So:
sqrt[3]{27000} = 30
Hardy Ramanujan Number Class 8 Important Questions
The chapter uses 1729 to show that numbers can hide surprising patterns. This makes cube numbers more interesting for students.
Taxicab Number Class 8 Questions
Q1. Why is 1729 called the Hardy-Ramanujan number?
1729 is called the Hardy-Ramanujan number because Ramanujan identified its special property.
It is the smallest number that can be written as the sum of two positive cubes in two different ways.
Q2. Write 1729 as the sum of two cubes in two ways.
1729 = 1^3 + 12^3
= 9^3 + 10^3
This makes 1729 the first taxicab number.
Q3. What is a taxicab number?
A taxicab number can be written as the sum of two positive cubes in two different ways.
The first taxicab number is 1729.
Q4. Express 4104 as the sum of two cubes in two ways.
4104 = 2^3 + 16^3
= 9^3 + 15^3
Q5. Express 13832 as the sum of two cubes in two ways.
13832 = 2^3 + 24^3
= 18^3 + 20^3
Class 8 Maths Chapter 1 Extra Questions with Answers
Class 8 maths chapter 1 extra questions help students practise mixed concepts from squares, cubes, roots and patterns.
Q1. Which number among (64^2), (108^2), (292^2), (36^2) has last digit 4?
A square ends in 4 when the number ends in 2 or 8.
108^2
ends in 4 because 8 squared gives units digit 4.
292^2
also ends in 4 because 2 squared gives units digit 4.
So, (108^2) and (292^2) have last digit 4.
Q2. Fill the blank: (4^2 + 5^2 + 20^2 = __^2).
4^2 + 5^2 + 20^2 = 16 + 25 + 400
= 441
441 = 21^2
So, the blank is 21.
Q3. Find cube roots of 4913, 12167 and 32768.
17^3 = 4913
23^3 = 12167
32^3 = 32768
So, the cube roots are 17, 23 and 32.
Q4. Find the smallest number by which 500 must be multiplied to make it a perfect cube.
Prime factorisation:
500 = 2^2 × 5^3
The factor 2 needs one more 2 to make a triplet.
So, multiply by 2.
500 × 2 = 1000
1000 = 10^3
Q5. Find the smallest number by which 156 must be multiplied to make it a perfect square.
Prime factorisation:
156 = 2^2 × 3 × 13
The factors 3 and 13 are unpaired.
So, multiply by:
3 × 13 = 39
156 × 39 = 6084
6084 = 78^2
Class 8 Maths Chapter 1 MCQs with Answers
Class 8 Maths Chapter 1 MCQs test quick recognition of square numbers, cube numbers, roots and special number patterns.
Q1. Which of the following is a perfect square?
(a) 2032
(b) 2048
(c) 1027
(d) 1089
Answer: (d) 1089
1089 = 33^2
Q2. Which number cannot be a perfect square?
(a) 625
(b) 1296
(c) 1537
(d) 2025
Answer: (c) 1537
It ends in 7, so it cannot be a perfect square.
Q3. The square root of 441 is:
(a) 19
(b) 20
(c) 21
(d) 22
Answer: (c) 21
21^2 = 441
Q4. Which of the following is a perfect cube?
(a) 9
(b) 64
(c) 500
(d) 156
Answer: (b) 64
64 = 4^3
Q5. The cube root of 729 is:
(a) 7
(b) 8
(c) 9
(d) 10
Answer: (c) 9
9^3 = 729
Q6. The smallest taxicab number is:
(a) 729
(b) 1000
(c) 1729
(d) 4104
Answer: (c) 1729
It can be written as the sum of two cubes in two different ways.
Competency-Based Questions on A Square and A Cube Class 8
These questions test whether students can apply squares, cubes and factor reasoning in new situations.
Q1. A number has factors (1, 2, 4, 8, 16). Will its locker remain open in the locker puzzle?
Yes, it will remain open.
The number has 5 factors, which is odd. This happens because 16 is a square number.
Q2. A square garden has area 625 m². Find its side.
Side of the square is:
√(625)
25^2 = 625
So, the side is 25 m.
Q3. A cube-shaped box contains 216 unit cubes. Find the side length.
The side length is:
sqrt[3]{216}
6^3 = 216
So, the side length is 6 units.
Q4. A student says 500 is a perfect cube because it ends in 0. Is this correct?
No, this is not correct.
A number ending in 0 is not always a perfect cube. A perfect cube must have prime factors in triplets.
500 = 2^2 × 5^3
The factor 2 is not in a triplet.
Q5. Why is prime factorisation useful for roots?
Prime factorisation shows whether factors can be grouped equally.
For square roots, factors must form pairs. For cube roots, factors must form triplets.
Class 8 Maths Important Questions Chapter-Wise