Stepwise Solutions & Important Questions for Class 8 Maths Chapter 1
Figure It Out (Pages 10-11)
1. Which of the following numbers are not perfect squares?
(i) 2032
(ii) 2048
(iii) 1027
(iv) 1089
Answer:
(i) 2032 → ends with 2 → not a perfect square.
(ii) 2048 → ends with 8 → not a perfect square.
(iii) 1027 → ends with 7 → not a perfect square.
(iv) 1089 → ends with 9; check: 33² = 1089 → is a perfect square.
Explanation: Squares can only end in 0, 1, 4, 5, 6, or 9. Any number ending in 2, 3, 7, or 8 cannot be a perfect square.
2. Which one among 64^2, 108^2, 292^2, 36^2 has the last digit 4?
Answer: 108² and 292².
64² = 4096 → last digit 6
108² = 11664 → last digit 4
292² = 85264 → last digit 4
36² = 1296 → last digit 6
Explanation: The last digit of a square depends only on the last digit of the base.
3. Given 125^2 = 15625, what is the value of 126^2?
Answer: 15876.
Work: Using (n+1)² = n² + 2n + 1 with n = 125:
126² = 15625 + 250 + 1 = 15876.
Explanation: Consecutive squares differ by 2n + 1.|
4. Find the length of the side of a square whose area is 441 m^2.
Answer: 21 m.
Explanation: If area = s², then s = √441 = 21.
5. Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.
Answer: 900.
Work: 4 = 2², 9 = 3², 10 = 2 × 5.
Smallest square divisible by all has even powers of all primes: 2² × 3² × 5² = 900.
Explanation: A perfect square needs even exponents in its prime factorisation.
6. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.
Answer: Multiply by 3; the product is 28224 and √28224 = 168.
Work: 9408 = 2⁶ × 3¹ × 7² → to make exponents even, multiply by one more 3.
Product: 9408 × 3 = 28224 = 2⁶ × 3² × 7² (a perfect square).
√28224 = 2³ × 3 × 7 = 8 × 3 × 7 = 168.
Explanation: Perfect squares have all prime exponents even.
7. How many numbers lie between the squares of the following numbers?
(i) 16 and 17
(ii) 99 and 100
Answer:
(i) 32
(ii) 198
Explanation: For consecutive integers n and n+1, count of integers strictly between n² and (n+1)² is (n+1)² − n² − 1 = 2n. So for n = 16, it’s 32; for n = 99, it’s 198.
8. In the following pattern, fill in the missing numbers:
12 + 22 + 22 = 32
22 + 32 + 62 = 72
32 + 42 + 122 = 132
42 + 52 + 202 = (__)2
92 + 102 + (__)2 = (__)2
Answer:
4² + 5² + 20² = 21²
9² + 10² + 90² = 91²
Explanation: The third term is the product of the two consecutive numbers (e.g., 4×5=20, 9×10=90), and the right side is one more than that product (20+1=21, 90+1=91), forming a Pythagorean-style identity.
9. How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.
Answer:
Big squares in a row = 9
Big squares in a column = 9
Tiny squares in each big square = 25
∴ Total number of tiny squares = 9 × 9 × 25 = 2025
Now, the prime factorization of 2025 = 3 × 3 × 3 × 3 × 5 × 5 = 452
Figure It Out (Pages 16-17)
1. Find the cube roots of 27000 and 10648.
Answer:
27000 = 27 × 1000 = (3 × 10)³ ⇒ ∛27000 = 30.
10648 = (2 × 11)³ ⇒ ∛10648 = 22.
Explanation: Break each number into a product of two perfect cubes. The cube root of a product is the product of the cube roots.
2. What number will you multiply by 1323 to make it a cube number?
Answer: Multiply by 7. The cube number obtained is 9261 (= 21³).
Work: 1323 = 3³ × 7². To make a perfect cube, every prime’s exponent must be a multiple of 3. Multiply by one more 7 to get 3³ × 7³.
3. State true or false. Explain your reasoning.
(i) The cube of any odd number is even.
(ii) There is no perfect cube that ends with 8.
(iii) The cube of a 2-digit number may be a 3-digit number.
(iv) The cube of a 2-digit number may have seven or more digits.
(v) Cube numbers have an odd number of factors.
Answer:
False. Odd × odd × odd stays odd (e.g., 3³ = 27).
False. 2³ = 8, 12³ = 1728 — cubes can end in 8.
False. The smallest 2-digit number is 10; 10³ = 1000 (4 digits). So a 2-digit cube cannot be 3-digit.
False. The largest 2-digit cube is 99³ = 970,299 (6 digits), not 7+ digits.
False (in general). Numbers have an odd number of factors only when they are perfect squares. Typical cubes like 8 (=2³) and 27 (=3³) have an even number of factors.
Explanation: Use basic parity rules, last-digit checks for cubes, and digit-count bounds: 10³ = 1000 (minimum for 2-digit bases) and 99³ = 970,299 (maximum for 2-digit bases). Factor-count parity is odd only for perfect squares.
4. You are told that 1331 is a perfect cube. Can you guess without factorisation what its cube root is? Similarly, guess the cube roots of 4913, 12167, and 32768.
Answer: ∛1331 = 11, ∛4913 = 17, ∛12167 = 23, ∛32768 = 32.
How to guess quickly:
Step 1 (range): Compare with nearby thousand-cubes: 10³=1000, 20³=8000, 30³=27000, 40³=64000, etc., to locate the tens digit.
Step 2 (units digit rule for cubes): The last digit of a perfect cube determines the last digit of the base uniquely (e.g., base ending 1→cube ends 1; 7→3; 3→7; 2→8; 8→2; 9→9; 0→0; 4→4; 5→5; 6→6).
Examples:
1331 lies between 10³ and 20³ and ends in 1 ⇒ base ends in 1 ⇒ 11.
4913 lies between 17³ (=4913) and 18³ (=5832); last digit 3 ⇒ base ends in 7 ⇒ 17.
12167 is between 20³ and 30³; last digit 7 ⇒ base ends in 3 ⇒ 23.
32768 is between 30³ and 40³; last digit 8 ⇒ base ends in 2 ⇒ 32.
5. Which of the following is the greatest? Explain your reasoning.
(i) 673 – 663
(ii) 433 – 423
(iii) 672 – 662
(iv) 432 – 422
Answer: (i) 673 – 663 is the greatest.
Reasoning (using identities):
For (i) and (ii): a³ − b³ = (a − b)(a² + ab + b²). Here, a − b = 1, so:
(i): 67³ − 66³ = 67² + 67·66 + 66² = 4489 + 4422 + 4356 = 13267.
(ii): 43³ − 42³ = 43² + 43·42 + 42² = 1849 + 1806 + 1764 = 5419.
For (iii) and (iv): a² − b² = (a − b)(a + b) with a − b = 1, so:
(iii): 67² − 66² = 67 + 66 = 133.
(iv): 43² − 42² = 43 + 42 = 85.
Comparing values: 13267 > 5419 > 133 > 85, hence (i) is greatest.
Understanding Squares and Cubes - NCERT Class 8 Maths Chapter 1 Solutions
Mastering the basics of square numbers and cube numbers in Class 8 Maths is key for exams. With step-by-step NCERT Solutions, you will gain clarity on perfect squares, cubes, and their properties for the academic year 2025-26.
To solve questions faster, remember that square roots and cube roots link directly to factorization. Use these patterns to identify numbers quickly and ace your school tests in NCERT Class 8 Chapter 1.
Practising exercise-based questions in “A Square and A Cube” will boost your understanding and confidence. Focus on understanding units digit patterns, cubes, and properties for a strong hold on this essential chapter.
FAQs on Class 8 Maths Chapter 1 NCERT Solutions: A Square and A Cube
1. What is cube in math class 8?
A cube in Class 8 Maths refers to the result of multiplying a number by itself three times.
• If a number is 'a', then its cube is a × a × a = a³.
• Example: The cube of 2 is 2 × 2 × 2 = 8.
• Cubes help in understanding the concept of perfect cubes and cube roots in NCERT Solutions for Class 8 Maths Chapter 1 A Square and A Cube.
• Mastering cubes is important for solving exercises and scoring well in the CBSE 2025–26 exams.
2. Which chapter is square and Square Roots class 8?
In Class 8 Maths NCERT, the chapter on Squares and Square Roots is Chapter 6.
• Chapter 1: A Square and A Cube covers basic concepts of squaring and cubing numbers.
• Chapter 6: Squares and Square Roots explores advanced properties, tricks, and root calculations.
• Both chapters are part of the CBSE Class 8 Mathematics Syllabus for session 2025-26.
• Practicing both chapters using NCERT Solutions for Class 8 Maths improves understanding and exam scores.
3. How do you explain squares and cubes?
Squares and cubes mean multiplying a number by itself two or three times, respectively.
• Square: Number × Number (n × n = n²).
• Cube: Number × Number × Number (n × n × n = n³).
• Example: Square of 3 is 9 (3²); Cube of 2 is 8 (2³).
• These operations are foundations in NCERT Solutions Class 8 Maths Chapter 1 A Square and A Cube.
• Mastery helps in faster calculations, problem-solving, and performing well in exams.
4. Are NCERT Solutions enough for Class 8 exams?
NCERT Solutions are usually sufficient for scoring full marks in Class 8 Maths exams.
• They cover all textbook exercises and important questions.
• Stepwise solutions match the CBSE marking scheme.
• Practicing these builds clarity and covers exam patterns.
• For higher-order thinking or extra practice, refer to Exemplar problems as well.
• Always revise using these solutions before exams for best results.
5. How can I write stepwise answers for Class 8 Maths Chapter 1 to score full marks?
To score full marks in Class 8 Maths Chapter 1 A Square and A Cube, write stepwise solutions as follows:
• Start with a clear statement of given data.
• Write relevant formulae or definitions.
• Show each calculation step neatly and sequentially.
• Box your final answer.
• Use keywords and language as used in the NCERT Solutions.
• Refer to accurate stepwise NCERT answers to match CBSE exam requirements.
6. Where can I find the NCERT Solutions PDF for Class 8 Maths Chapter 1?
You can download the Class 8 Maths Chapter 1 NCERT Solutions PDF from trusted educational websites.
• Look for platforms offering free, teacher-reviewed PDFs.
• Ensure the content aligns with the CBSE 2025–26 syllabus.
• Use these PDFs for offline revision and exam preparation.
7. Do examiners award partial marks for correct steps even if the final answer is wrong?
Yes, in CBSE Maths exams, partial marks are often awarded for correct steps even if the final answer is incorrect.
• Always show stepwise calculations as in NCERT Solutions for Class 8 Maths Chapter 1.
• Follow the CBSE marking scheme by explaining logic and formula usage.
• This helps you maximise your score, even with minor calculation errors.
8. What are the most important topics from Class 8 Maths Chapter 1 – A Square and A Cube?
The most important topics in Class 8 Maths Chapter 1 include:
• Finding squares and cubes of numbers
• Properties of square and cube numbers
• Patterns involving squares and cubes
• Concept of perfect squares and perfect cubes
• Applications in word problems
• Stepwise problem solving as per NCERT Solutions
9. Are diagrams or definitions mandatory in answers for Chapter 1?
Including definitions and diagrams (where applicable) is recommended for full marks in CBSE Maths exams.
• Definitions clarify your understanding (e.g., what is a square number or cube number).
• Diagrams are rarely required in Chapter 1, but clear formatting and boxed answers help.
• Always follow the stepwise method in your NCERT Solutions to meet exam expectations.
10. How do I revise Class 8 Maths Chapter 1 efficiently before exams?
To quickly revise Class 8 Maths Chapter 1 A Square and A Cube:
• Go through key definitions and formulae first.
• Practice stepwise NCERT Solutions, focusing on important questions.
• Use quick revision notes and flashcards.
• Attempt sample papers and MCQs for extra practice.
• Clarify doubts using solution PDFs and revision planners.
