Square Roots and Cube Roots: Tables, Shortcut Tricks and Practice MCQs

Square roots and cube roots show up constantly in simplification questions in SSC, railway and banking exams. If you memorise a small set of squares and cubes and learn one unit-digit trick, most of these become 10-second questions.

Squares You Should Know (1–30)

n	n²	n	n²	n	n²
1	1	11	121	21	441
2	4	12	144	22	484
3	9	13	169	23	529
4	16	14	196	24	576
5	25	15	225	25	625
6	36	16	256	26	676
7	49	17	289	27	729
8	64	18	324	28	784
9	81	19	361	29	841
10	100	20	400	30	900

Cubes You Should Know (1–15)

n	n³	n	n³	n	n³
1	1	6	216	11	1331
2	8	7	343	12	1728
3	27	8	512	13	2197
4	64	9	729	14	2744
5	125	10	1000	15	3375

The Unit-Digit Trick for Square Roots

A perfect square can only end in 0, 1, 4, 5, 6 or 9 (never 2, 3, 7 or 8). To find the root of a 4-digit perfect square:

1. Split off the last two digits. The remaining part tells you the tens digit: find the largest square ≤ that part.
2. The unit digit of the square tells you the two candidates for the root's unit digit: 1→(1,9), 4→(2,8), 5→(5), 6→(4,6), 9→(3,7), 0→(0).
3. Pick between the two candidates by comparing with the square of the "middle" number ending in 5.

Example: √4489. The part before the last two digits is 44; 6² = 36 ≤ 44 < 49, so the tens digit is 6. It ends in 9, so the root ends in 3 or 7 — candidates 63 or 67. Since 65² = 4225 < 4489, the answer is 67.

The Unit-Digit Trick for Cube Roots

Cube unit digits are unique: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0 (2 and 8 swap; 3 and 7 swap). For a cube up to 6 digits: the last digit gives the root's unit digit, and the part before the last three digits gives the tens digit (largest cube ≤ it).

Example: ∛12167. Ends in 7 → root ends in 3. Before the last three digits: 12; 2³ = 8 ≤ 12 < 27, so tens digit is 2. Answer: 23.

Practice MCQs

Q1. √2025 = ?

(a) 35 (b) 45 (c) 55 (d) 65 — Answer: (b) 45. Ends in 5 → root ends in 5; 4 before last two digits → tens digit 4.

Q2. √1764 = ?

(a) 38 (b) 42 (c) 44 (d) 48 — Answer: (b) 42. 17 → tens digit 4; ends in 4 → 2 or 8; 45² = 2025 > 1764 → 42.

Q3. √3249 = ?

(a) 53 (b) 57 (c) 63 (d) 67 — Answer: (b) 57. 32 → tens 5; ends in 9 → 3 or 7; 55² = 3025 < 3249 → 57.

Q4. √2916 = ?

(a) 54 (b) 56 (c) 44 (d) 64 — Answer: (a) 54. 29 → tens 5; ends in 6 → 4 or 6; 55² = 3025 > 2916 → 54.

Q5. √5329 = ?

(a) 67 (b) 77 (c) 73 (d) 83 — Answer: (c) 73. 53 → tens 7; ends in 9 → 3 or 7; 75² = 5625 > 5329 → 73.

Q6. √6084 = ?

(a) 74 (b) 78 (c) 82 (d) 86 — Answer: (b) 78. 60 → tens 7; ends in 4 → 2 or 8; 75² = 5625 < 6084 → 78.

Q7. √1444 = ?

(a) 32 (b) 34 (c) 36 (d) 38 — Answer: (d) 38. 14 → tens 3; ends in 4 → 2 or 8; 35² = 1225 < 1444 → 38.

Q8. √1024 = ?

(a) 28 (b) 32 (c) 34 (d) 36 — Answer: (b) 32. 10 → tens 3; ends in 4 → 2 or 8; 35² = 1225 > 1024 → 32.

Q9. ∛1728 = ?

(a) 12 (b) 14 (c) 16 (d) 18 — Answer: (a) 12. Ends in 8 → root ends in 2; 1 before last three digits → tens 1.

Q10. ∛2744 = ?

(a) 12 (b) 14 (c) 16 (d) 18 — Answer: (b) 14. Ends in 4 → ends in 4; 2 → tens 1.

Q11. ∛4913 = ?

(a) 13 (b) 15 (c) 17 (d) 19 — Answer: (c) 17. Ends in 3 → root ends in 7; 4 → tens 1.

Q12. ∛9261 = ?

(a) 19 (b) 21 (c) 23 (d) 27 — Answer: (b) 21. Ends in 1 → root ends in 1; 9 → tens 2.

How to Use This in the Exam

Simplification questions rarely say "find the square root" directly — the roots hide inside larger expressions. Memorise the tables above, apply the unit-digit tricks, and practise under time with Pareeksha's topic practice sets and the quant sections of our mock tests.

FAQs

Can a perfect square end in 2, 3, 7 or 8? No — that is the fastest way to eliminate options.

Does the cube-root trick work for non-perfect cubes? No. Both unit-digit tricks assume the number is a perfect square or cube, which exam questions state or imply.