CBSE Class 8 Mathematics A Square and A Cube Very Short Answer Questions with Answers

Very Short Answer Questions

Question 1. What is a square number? Give one example.
Answer:
A number that can be written as the product of a number with itself is a square number. For example, 25 = 5 × 5 = 5².

Teacher's Note:
1. The name comes from the area of a square.
2. Any example like 36 = 6² is fine.

Question 2. Write the first five perfect cubes.
Answer:
1, 8, 27, 64 and 125, that is 1³, 2³, 3³, 4³ and 5³.

Teacher's Note:
1. Cube means the number multiplied by itself three times.
2. Learn the cubes up to 10³ = 1000.

Question 3. Which digits can never appear in the units place of a perfect square?
Answer:
The digits 2, 3, 7 and 8 never appear there; a square ends only in 0, 1, 4, 5, 6 or 9.

Teacher's Note:
1. This is a quick test to reject non-squares.
2. It can never prove that a number is a square.

Question 4. In the locker puzzle of Queen Ratnamanjuri, write all the locker numbers that remain open at the end.
Answer:
1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 — the square numbers up to 100, because only they have an odd number of factors.

Teacher's Note:
1. Each locker is toggled once for each factor of its number.
2. An odd number of toggles leaves a locker open.

Question 5. How many unit cubes of edge 1 cm are needed to build a cube of edge 3 cm?
Answer:
The cube has 3 layers of 3 × 3 unit cubes, so 3³ = 27 unit cubes are needed.

Teacher's Note:
1. Think in layers: cubes in one layer × number of layers.
2. Edge n needs n³ unit cubes.

Question 6. Find the value of √169.
Answer:
13 × 13 = 169, so √169 = 13.

Teacher's Note:
1. 169 ends in 9, so the root ends in 3 or 7.
2. Squares from 11² to 20² are worth memorising.

Question 7. Write 1 + 3 + 5 + 7 + 9 + 11 as the square of a number.
Answer:
The sum has six odd numbers starting from 1, so it equals 6² = 36.

Teacher's Note:
1. The sum of the first n odd numbers is n².
2. Count the terms instead of adding them.

Question 8. Write both the integral square roots of 144.
Answer:
12 and −12, since 12² = 144 and (−12)² = 144.

Teacher's Note:
1. A negative number times itself is positive.
2. The symbol √ shows only the positive root.

Question 9. Find ∛1000.
Answer:
10 × 10 × 10 = 1000, so ∛1000 = 10.

Teacher's Note:
1. 1000 has three zeros, one group of three.
2. ∛ asks for a number used three times.

Question 10. Write the first five triangular numbers.
Answer:
1, 3, 6, 10 and 15.

Teacher's Note:
1. Each one adds the next counting number: 1, 1 + 2, 1 + 2 + 3, …
2. Two consecutive ones add to a square.

Question 11. Is the square of an odd number odd or even? Give one example.
Answer:
The square of an odd number is always odd; for example 7² = 49.

Teacher's Note:
1. Odd × odd is always odd.
2. Similarly, even × even is even.

Question 12. Who compiled the first known lists of perfect squares and perfect cubes, and roughly when?
Answer:
The Babylonians, on clay tablets, around 1700 BCE.

Teacher's Note:
1. They used these lists for land measurement and architecture.
2. BCE means before the Common Era.

Question 13. Without calculating, state whether 2032 is a perfect square and give the reason.
Answer:
2032 is not a perfect square, because it ends in 2 and no perfect square ends in 2, 3, 7 or 8.

Teacher's Note:
1. Check the units digit first.
2. This test rules numbers out instantly.

Question 14. Given that 35² = 1225, use the odd-number pattern to find 36².
Answer:
36² = 35² + the 36th odd number. The 36th odd number is 2 × 36 − 1 = 71.
So 36² = 1225 + 71 = 1296.

Teacher's Note:
1. (n + 1)² = n² + 2n + 1.
2. Check: 36 × 36 = 1296.

Question 15. How many natural numbers lie between 20² and 21²?
Answer:
20² = 400 and 21² = 441, so there are 441 − 400 − 1 = 40 numbers between them (rule 2n with n = 20).

Teacher's Note:
1. Between n² and (n + 1)² lie 2n numbers.
2. Leave out both end squares.

Question 16. Find √324 using prime factorisation.
Answer:
324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 3 × 3) × (2 × 3 × 3) = 18², so √324 = 18.

Teacher's Note:
1. Split the prime factors into two equal groups.
2. One group gives the square root.

Question 17. Is 500 a perfect cube? Give the reason using its prime factorisation.
Answer:
No. 500 = 2 × 2 × 5 × 5 × 5. The factor 2 occurs only twice, so the factors cannot be split into three identical groups.

Teacher's Note:
1. Every prime must appear in groups of three.
2. 500 would need another 2 to become 1000 = 10³.

Question 18. How many zeros will 600² have at the end?
Answer:
600² = 360000, which ends with 4 zeros — double the 2 zeros of 600.

Teacher's Note:
1. Squaring doubles the number of end zeros.
2. 6² = 36, then attach four zeros.

Question 19. Find the cube root of 3375 by prime factorisation.
Answer:
3375 = 3 × 3 × 3 × 5 × 5 × 5 = (3 × 5)³ = 15³, so ∛3375 = 15.

Teacher's Note:
1. Take one factor from each triplet.
2. 3375 ends in 5, so the cube root ends in 5.

Question 20. Express 1729 as the sum of two cubes in two different ways.
Answer:
1729 = 1³ + 12³ = 1 + 1728 and 1729 = 9³ + 10³ = 729 + 1000.

Teacher's Note:
1. This is the famous Hardy–Ramanujan number.
2. It is the smallest number with this property.

Question 21. Write the sum 21 + 23 + 25 + 27 + 29 as the cube of a number.
Answer:
These are 5 consecutive odd numbers with sum 125 = 5³.
This follows the pattern that the nth cube is a sum of n consecutive odd numbers.

Teacher's Note:
1. The middle number is 25, so the sum is 5 × 25.
2. 5 × 25 = 125 = 5³.

Question 22. Between which two consecutive whole numbers does √250 lie, and which of the two is it nearer to?
Answer:
15² = 225 and 16² = 256, so √250 lies between 15 and 16. Since 250 is much nearer to 256, it is nearer to 16.

Teacher's Note:
1. Trap the number between two consecutive squares.
2. 250 is 6 below 256 but 25 above 225.

Question 23. In the table of successive differences for perfect squares, after how many levels do all the differences become equal, and what is that common value?
Answer:
They become equal after 2 levels, and the common difference is 2.

Teacher's Note:
1. Level 1 gives the odd numbers 3, 5, 7, …
2. Their differences are all 2.

Question 24. What do the Sanskrit terms varga-mula and ghana-mula stand for, and what does the word mula mean?
Answer:
Varga-mula means square root and ghana-mula means cube root. Mula means the root of a plant, and so the basis or origin.

Teacher's Note:
1. The word 'root' in maths comes from this idea.
2. It passed into Arabic as jidhr and Latin as radix.

Question 25. Find the smallest square number that is divisible by each of 6, 9 and 15.
Answer:
LCM of 6, 9 and 15 = 90 = 2 × 3 × 3 × 5. The single 2 and single 5 need partners, so multiply by 2 × 5 = 10.
The smallest such square is 900 = 30².

Teacher's Note:
1. Start from the LCM, then complete the pairs.
2. Multiply by the unpaired primes only.

Question 26. Find the smallest number by which 2800 must be multiplied so that the product is a perfect square, and write the square root of that product.
Answer:
2800 = (2 × 2) × (2 × 2) × (5 × 5) × 7; only 7 is unpaired, so multiply by 7.
2800 × 7 = 19600 = 140², so the square root is 140.

Teacher's Note:
1. Pair up the prime factors and find the odd one out.
2. Check: 140 × 140 = 19600.

Question 27. Can a perfect cube end with exactly two zeros? Give a reason.
Answer:
No. End zeros of a cube come from factors of 10 in triplets, so a cube ends with 3, 6, 9, … zeros, never exactly two.

Teacher's Note:
1. Cubing triples the number of end zeros.
2. 10³ = 1000 has three zeros.

Question 28. A natural number has exactly 15 factors. Must it be a perfect square? Explain.
Answer:
Yes. Factors come in partner pairs, so the count is odd only when one pair has equal partners, that is, only for a perfect square.
For example, 324 = 18² has exactly 15 factors.

Teacher's Note:
1. Odd number of factors means a square number.
2. 15 is odd, so the number must be a square.

Question 29. Find the perfect square that lies between 2000 and 2100.
Answer:
44² = 1936 and 46² = 2116 fall outside, while 45² = 2025 lies between 2000 and 2100. So the square is 2025.

Teacher's Note:
1. Use known squares near the range.
2. 45² = 2025 is worth remembering.

Question 30. Explain briefly why the number of zeros at the end of a perfect square is always even.
Answer:
If a number ends with k zeros, its square ends with 2k zeros, because each factor of 10 appears twice in the square. Since 2k is always even, a square has an even number of end zeros.

Teacher's Note:
1. Example: 700² = 490000 has 4 zeros.
2. A number with 1, 3 or 5 end zeros cannot be a square.