Prelims · 8 Mental Ability · 8.1 Number system and simplification

8.1.7 Squares, square roots, cubes and cube roots: fast methods and estimation

Factorise to find exact roots, use the last digit and the leading group to read roots of large perfect squares and cubes in seconds, and estimate the rest with a ± b/(2a).

  • APPSC Group 2 · Prelims
  • Standard level
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Core facts

  • Learn squares to 30 and cubes to 15. Useful ones: 13² = 169, 17² = 289, 19² = 361, 23² = 529, 27² = 729; 11³ = 1,331, 12³ = 1,728, 13³ = 2,197, 14³ = 2,744, 15³ = 3,375.
  • Square root by factorisation: pair the prime factors; take one from each pair. Cube root: group in threes.
  • A perfect square has every prime exponent even; a perfect cube has every exponent a multiple of 3.
  • Last digit of squares: only 0, 1, 4, 5, 6, 9. A square never ends in 2, 3, 7 or 8, or in an odd number of zeros.
Square ends inRoot ends in
11 or 9
42 or 8
93 or 7
64 or 6
55
  • Cubes keep the last digit for 0, 1, 4, 5, 6, 9 and swap 2 ↔ 8 and 3 ↔ 7.
  • Digits: a square of an n-digit number has 2n − 1 or 2n digits. A root of a 2k-digit or (2k − 1)-digit square has k digits.
  • Roots of decimals: the number of decimal places must be even (square) or a multiple of 3 (cube); the root has half (or a third) as many: √0.0169 = 0.13.

Example 1. Find √5,184. Solution. 5,184 = 2⁶ × 3⁴. Root = 2³ × 3² = 8 × 9. Answer: 72

Example 2. Find ∛2,744. Solution. 2,744 = 2³ × 7³. Root = 2 × 7. Answer: 14

Example 3. Least number by which 1,176 must be multiplied to make a perfect square. Solution. 1,176 = 2³ × 3 × 7². Make every exponent even: multiply by 2 × 3. Answer: 6

Explained

Type 1: square root of a large perfect square (no long division). Drop the last two digits; find the largest square at or below what is left; that gives the leading part. The last digit gives two choices; pick by comparing with n(n + 1).

Example 4. Find √119,716. Solution. Last digit 6, so the root ends in 4 or 6. Drop "16": 1,197. 34² = 1,156 ≤ 1,197 < 35² = 1,225, so the root is 34_. 34 × 35 = 1,190 < 1,197, so take the larger ending, 6. Answer: 346

Type 2: cube root of a perfect cube. Last digit by the swap rule; drop the last three digits and find the largest cube at or below the rest.

Example 5. Find ∛438,976. Solution. Ends in 6, so the root ends in 6. Drop "976": 438. 7³ = 343 ≤ 438 < 8³ = 512. Answer: 76

Type 3: roots of decimals.

Example 6. Find √0.000441. Solution. Six decimal places, so the root has three. √441 = 21, giving 0.021. Answer: 0.021

Type 4: estimation. √(a² + b) ≈ a + b/(2a) for small b.

Example 7. Estimate √83 to two decimal places. (a) 9.06 (b) 9.11 (c) 9.16 (d) 9.21 Solution. 83 = 81 + 2: 9 + 2/18 ≈ 9.11. (True value 9.110.) Answer: (b) 9.11

Example 8. Given √5 = 2.236, find 1/√5. Solution. 1/√5 = √5/5 = 2.236 ÷ 5. Answer: 0.4472

  • Always rationalise before dividing; dividing 1 by 2.236 is slow.

Type 5: least number to add or subtract.

Example 9. (i) Least number to subtract from 6,000 to get a perfect square. (ii) Least number to add. Solution. 77² = 5,929 and 78² = 6,084. (i) 6,000 − 5,929 = 71. (ii) 6,084 − 6,000 = 84. Answer: (i) 71 (ii) 84

Type 6: spot the perfect square. Use the last digit, then the tens digit, then the digital root.

Example 10. Only one is a perfect square: (a) 6,246 (b) 6,241 (c) 6,248 (d) 6,251 Solution. (c) ends in 8: rejected. A square ending in 6 has an odd tens digit, so (a) is out. A square ending in 1 has an even tens digit, so (d) is out. 79² = 6,241. Answer: (b) 6,241

Speed tips.

  • Squares ending in 1, 4, 9 or 5 have an even tens digit (25 for 5); squares ending in 6 have an odd tens digit.
  • Digital root (repeated digit sum) of a square is 1, 4, 7 or 9.

Common traps

  • ✗ "√0.0169 = 0.013." ✓ Four decimal places give two in the root: 0.13.
  • ✗ "6,246 ends in 6, so it may be a square." ✓ A square ending in 6 has an odd tens digit; 6,246 is not a square.
  • ✗ "The cube root of 438,976 ends in 4." ✓ Cubes keep 6 → root ends in 6 (76). Only 2 ↔ 8 and 3 ↔ 7 swap.
  • ✗ "Squares or cubes up to 1,000 = 31 + 10 = 41." ✓ Subtract the 3 sixth powers counted twice: 38.
  • ✗ "To make 1,176 a square, multiply by 3." ✓ The 2 also has an odd exponent (2³): multiply by 6.

Quick revision

  • Squares end only in 0, 1, 4, 5, 6, 9.
  • Root ends: 1 → 1/9, 4 → 2/8, 9 → 3/7, 6 → 4/6, 5 → 5.
  • Cube roots: 2 ↔ 8 and 3 ↔ 7 swap; others stay.
  • Large square: drop two digits, find the leading square, choose with n(n + 1).
  • Large cube: drop three digits, find the leading cube.
  • √(a² + b) ≈ a + b/(2a).
  • Square: all exponents even. Cube: all multiples of 3.
  • Tens digit odd only for squares ending in 6.
  • Digital root of a square: 1, 4, 7, 9.
  • √(n + √(n + …)) = k when n = k(k − 1).
Go deeper: Deeper (beyond what APPSC Group 2 usually asks)

Deeper Advanced

Example 11. Find √(248 + √(52 + √144)). Solution. Work from inside: √144 = 12; √(52 + 12) = √64 = 8; √(248 + 8) = √256. Answer: 16

Example 12. Find x = √(12 + √(12 + √(12 + …))) (continuing for ever). Solution. The expression under the first root contains x again: x = √(12 + x), so x² − x − 12 = 0, (x − 4)(x + 3) = 0. x is positive. Answer: 4

  • General: √(n + √(n + …)) = k when n = k(k − 1): 2, 6, 12, 20, 30 give 2, 3, 4, 5, 6.

Example 13. How many numbers from 1 to 1,000 are neither perfect squares nor perfect cubes? (a) 959 (b) 962 (c) 969 (d) 972 Solution. Squares: 31 (31² = 961). Cubes: 10. Both (sixth powers 1, 64, 729): 3. Squares or cubes = 31 + 10 − 3 = 38. 1,000 − 38. Answer: (b) 962

Example 14. A commander has 2,000 soldiers and arranges as many as possible in a solid square. How many are left out? Solution. 44² = 1,936 ≤ 2,000 < 45² = 2,025. Left = 2,000 − 1,936. Answer: 64

Example 15. A number's square has 7 digits. How many digits does the number have? Solution. 1,000² = 1,000,000 (7 digits) and 3,162² = 9,998,244 (7 digits) while 3,163² has 8. All such numbers lie from 1,000 to 3,162. Answer: 4

  • Squares of numbers ending in 5: n5² = n(n + 1) followed by 25: 65² = 4,225. (More in MAB-NS-13.)
  • Product under roots: √a × √b = √(ab): √2 × √3 = √6 ≈ 2.449.

Sources: NCERT Mathematics textbook, Class 8 (squares and square roots; cubes and cube roots); Standard properties of perfect squares and cubes (last digits, digital roots)

Last checked 2026-10-17.

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