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Squares and square roots come back in almost every later chapter: Pythagoras, areas, quadratic equations and the formulas of physics. The new Class 8 book teaches them through patterns rather than rules, which makes them easier to remember if you work through the patterns yourself. This guide covers the patterns, three methods for square roots and the mistakes to watch for.
Where is this topic in the 2026-27 NCERT books?
It is the first section of Ganita Prakash Grade 8 Part I, Chapter 1, "A Square and A Cube". The older Class 8 chapter "Squares and Square Roots" covered the same ground, with the long division method for square roots and Pythagorean triplets added. The new chapter opens with a puzzle about 100 lockers, toggled by 100 people, and asks which stay open. The answer is the lockers with square numbers. Why? A locker is toggled once for each factor of its number, factors come in pairs, and only a square has a factor paired with itself, so only squares have an odd number of factors.
What patterns do square numbers follow?
Ganita Prakash asks you to fill in the squares of 1 to 30 and look for patterns. The main ones:
Last digits. Squares end only in 0, 1, 4, 5, 6 or 9. So a number ending in 2, 3, 7 or 8 cannot be a square. The reverse is not true: 26 ends in 6 but is not a square.
Zeros. A square ends in an even number of zeros: 100, 40000, 490000.
Odd numbers. 1 = 1², 1 + 3 = 2², 1 + 3 + 5 = 3². In general, the sum of the first n odd numbers is n². Since the nth odd number is 2n − 1, you can get the next square by adding it: 35² = 1225, so 36² = 1225 + 71 = 1296.
Gaps. Between n² and (n + 1)² there are 2n numbers that are not squares, as the older chapter shows. Between 16 and 25, for example, there are 2 × 4 = 8.
How do you find a square root? Three methods
Method 1: subtract odd numbers. Take 81 and subtract 1, 3, 5, 7, and so on. You reach 0 after nine steps, so √81 = 9. Good for small numbers, slow for big ones.
Method 2: pair the prime factors. This is the method to master.
Worked example: is 2025 a perfect square? 2025 = 3 × 3 × 3 × 3 × 5 × 5. Pair them: (3 × 3) × (3 × 3) × (5 × 5). Every factor has a partner. Take one from each pair: 3 × 3 × 5 = 45. So √2025 = 45. Check: 45 × 45 = 2025.
Worked example: what is the smallest number you must multiply 1008 by to get a perfect square? 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7. The 2s and 3s pair up, but 7 has no partner. Multiply by 7. 1008 × 7 = 7056 = (2 × 2 × 3 × 7)² = 84².
Method 3: estimate, then narrow down. Ganita Prakash finds √1936 like this. 1936 lies between 1600 (40²) and 2500 (50²), so the root is between 40 and 50. It ends in 6, so the root ends in 4 or 6: 44 or 46. Since 45² = 2025 is more than 1936, the root is below 45. So √1936 = 44.
For a number that is not a square, the same idea gives an estimate. The book's example: 15² = 225 and 16² = 256, so √250 is between 15 and 16, and closer to 16.
How can a square help you calculate faster?
Splitting a number makes mental squaring easy, using (a + b)² = a² + 2ab + b², which you meet properly in the chapter on algebraic identities.
51² = (50 + 1)² = 2500 + 100 + 1 = 2601. 65² = (60 + 5)² = 3600 + 600 + 25 = 4225 (the book's own example).
What mistakes should you avoid?
Taking a square root term by term. √(9 + 16) = √25 = 5, not √9 + √16 = 7.
Leaving out a factor when pairing. In 1008, students pair the 2s and 3s and forget the 7, then claim it is a square.
Forgetting the negative root. Ganita Prakash notes that 64 has two square roots, +8 and −8, though the chapter then works only with the positive one. Read the question.
Confusing doubling with squaring. 5² is 25, not 10.
How should you practise?
Squares reward short, regular practice. A large review of spacing studies, Cepeda and colleagues (2006), pooled 839 assessments of the "distributed practice effect": the finding that study spread over separate sessions is remembered better than the same study crammed into one. So learn the squares a few at a time over several days, not all at once the night before.
Learn squares to 30 in three batches, 1 to 10, 11 to 20, 21 to 30, a few days apart. Quiz yourself from a blank page.
Two factor problems a day: one "is this a square?" and one "smallest multiplier".
One estimate a day: pick a four-digit square and find its root by the narrowing method before checking.
Mix with cubes from the same chapter once a week; our guide on cubes and cube roots in Class 8 has the matching methods.
If prime factorisation is where you get stuck, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer, so you can see each step of the pairing.
Right now, find √3136 by estimation, then check it by prime factors.
