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Class 8 Maths: A Square and A Cube, 8 practice questions with answers

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These are 8 practice questions on Chapter 1, A Square and A Cube, of the new NCERT Class 8 Maths textbook, Ganita Prakash, Part 1. Each has a worked answer and the textbook page it comes from. They cover judging a claim from the units digit, testing a square by prime factorisation, the odd numbers that add up to a square, narrowing down a square root, testing one number for both a cube and a square, the units digit of a cube, squares and cubes of a decimal less than 1, and the area of a square field from its fencing. Try each one on paper before you open the answer. The next chapter, Power Play, has the laws of exponents: Class 8 Power Play (Exponents) questions.

These questions are our own, drafted with AI help. Every number was re-computed by a test script and none of the numbers is taken from the chapter, but no practising teacher has reviewed them yet. If you find a mistake, please tell us.

What the chapter says you need

Facts from the NCERT textbook (Ganita Prakash, Grade 8, Part 1, Chapter 1, printed pages 1 to 17):

Squares

Question 1. Meena says, "7396 ends in 6, so it must be a perfect square." Ravi says, "9218 ends in 8, so it cannot be a perfect square." Whose reasoning is correct? Is 7396 a perfect square?

Answer: Ravi's reasoning is correct; Meena's is not, although 7396 does happen to be a square (86²). No perfect square ends in 2, 3, 7 or 8, so 9218 is certainly not a square. But ending in 0, 1, 4, 5, 6 or 9 does not make a number a square: 7406 also ends in 6, yet it lies between 86² = 7396 and 87² = 7569, so it is not a square. Meena's reason proves nothing; we have to check, and 86 × 86 = 7396, so 7396 is a perfect square. (Units digits of squares, page 4.)

Question 2. 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7. Use this to show that 7056 is a perfect square and find its square root. Is 2 × 7056 = 14112 also a perfect square?

Answer: √7056 = 84; 14112 is not a perfect square. The factors of 7056 split into two identical groups, (2 × 2 × 3 × 7) × (2 × 2 × 3 × 7), so 7056 = (2 × 2 × 3 × 7)² = 84². Multiplying by 2 adds a fifth 2: 14112 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7. The extra 2 has no partner, so the factors cannot be split into two identical groups, and 14112 is not a perfect square. (Perfect squares by prime factorisation, page 9.)

Question 3. 3364 is a perfect square. If you add up odd numbers starting from 1 (1 + 3 + 5 + …) until the total is 3364, how many odd numbers do you add, and what is the last one?

Answer: 58 odd numbers; the last one is 115. The sum of the first n odd numbers is n². Since 58 × 58 = 3364, 3364 is the sum of the first 58 odd numbers. The nth odd number is 2n − 1, so the 58th is 2 × 58 − 1 = 115. (Squares and odd numbers, page 6.)

Question 4. √2967 lies between 54 and 55, because 54² = 2916 and 55² = 3025. Work out 54.5² and use it to decide whether √2967 is nearer 54 or 55.

Answer: nearer 54. 54.5² = 54.5 × 54.5 = 2970.25. Since 2967 < 2970.25, √2967 is less than 54.5, the number midway between 54 and 55, so √2967 is nearer 54. (Halving the interval when estimating a square root, page 9; estimating square roots, page 10.)

Cubes, decimals and a field

Question 5. 21952 = 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 7. Is 21952 a perfect cube? Is it a perfect square? Give the root where there is one.

Answer: a perfect cube, with cube root 28; not a perfect square. The factors split into three identical groups, (2 × 2 × 7) × (2 × 2 × 7) × (2 × 2 × 7), so 21952 = (2 × 2 × 7)³ = 28³. To be a square they would have to split into two identical groups, but there are three 7s, and an odd number of 7s cannot be shared equally between two groups, so 21952 is not a perfect square. (Perfect cubes by prime factorisation, page 14; perfect squares by prime factorisation, page 9.)

Question 6. Without working out the whole cubes, find the units digit of 47³ and of 58³. Then check your answers.

Answer: 47³ ends in 3; 58³ ends in 2. When a number is multiplied by itself three times, only the units digits decide the units digit of the result. 47 ends in 7, and 7 × 7 × 7 ends in 3, so 47³ ends in 3. 58 ends in 8, and 8 × 8 × 8 ends in 2, so 58³ ends in 2. Check: 47³ = 47 × 47 × 47 = 103823 and 58³ = 58 × 58 × 58 = 195112. (Cubes and their last digits, pages 12 and 13.)

Question 7. Find (0.47)² and (0.47)³. Which is larger, and why?

Answer: (0.47)² = 0.2209 is larger than (0.47)³ = 0.103823. (0.47)² = 0.47 × 0.47 = 0.2209, and (0.47)³ = 0.2209 × 0.47 = 0.103823. Multiplying a positive number by 0.47, which is less than 1, makes it smaller, so the cube is smaller than the square. (Squares of decimals, page 3; cubes of decimals, page 13.)

Question 8. A square field needs 188 m of fencing to go once around it. What is its area?

Answer: 2209 m². The four sides are equal, so each side is 188 ÷ 4 = 47 m. Area = 47 × 47 = 2209 m². (Area of a square as side × side, page 3.)

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Sources

All opened and read on 22 October 2026. Page numbers are the printed page numbers of the textbook; printed page = PDF page + 0.

The rules are from the textbook pages above. The questions, worked answers and checks are our own, and every number was re-computed by script. This guide is general information; follow your teacher and the textbook.