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NCERT Solutions Ganita Prakash (Part 1) Chapter 1 Square Roots — In-text Questions

Book page 8 Updated on2026-09-05

Q1.
What is the square root of 64?
Answer

8 × 8 = 64, and (–8) × (–8) = 64 as well.

82 = 64 and (–8)2 = 64
So the square roots of 64 are +8 and –8
√64 = ±8; in this chapter we take √64 = 8
Why there are two: A negative times a negative is positive, so any perfect square has two integer roots that differ only in sign. In general √(n2) = ±n. The symbol √ by itself is reserved for the positive one, which is why √100 = 10, not –10.
Q2.
Given a number, such as 576 or 327, how do we find out if it is a perfect square? If it is a perfect square, how can we find its square root?
Answer

Use the tests in order — cheapest first.

  1. Units digit. 327 ends in 7, so it is certainly not a perfect square. 576 ends in 6, so it survives this test but is not yet settled.
  2. List the squares. 202 = 400, 212 = 441, 222 = 484, 232 = 529, 242 = 576. So 576 is a perfect square and √576 = 24. This works, but it is slow for large numbers.
  3. Subtract odd numbers. Take away 1, 3, 5, … in turn; landing exactly on 0 at step k means the number is k2.
  4. Prime factorisation. Split the prime factors into two identical groups. 576 = 26 × 32 = (23 × 3) × (23 × 3) = 24 × 24.
576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
= (2 × 2 × 2 × 3) × (2 × 2 × 2 × 3)
= 242, so √576 = 24
Why prime factorisation is the reliable test: If a number is m2, every prime in m appears twice as often in m2. So a perfect square is exactly a number in which every prime occurs an even number of times — and halving each exponent reads off the root directly.
Q3.
Can we find the square root of 729 using this method?
Answer

Yes, but it is time-consuming — it takes 27 subtractions.

729 – 1 = 728, 728 – 3 = 725, 725 – 5 = 720, …
… and the 27th subtraction, of 53, leaves 0
So 729 = 272 and √729 = 27
Why it is slow: Finding √N this way needs about √N steps. For 81 that is 9 steps, which is fine; for 729 it is 27, and for a six-digit number it would run into the hundreds. Prime factorisation is far shorter: 729 = 36 = (33)2 = 272, done in one line.
Check it yourself: The odd numbers 1 + 3 + 5 + … + 53 add to 272 = 729, because 53 = 2 × 27 – 1.
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