Squares, cubes and roots in seconds
Almost every square you need is one of four patterns: ends in 5, near 50, near 100, or (a+b)2. Roots of perfect squares and cubes come from the last digit plus the leading block.
Why this topic matters
Squares and roots appear inside mensuration, number system, DI and simplification. Knowing four square patterns and one root trick saves a minute every time.
Questions like 'square root of 7744' or 'value of 1132' are asked directly, and squares hide inside Pythagoras, areas and compound interest. Learn squares to 30 by heart (cheat sheet) and use the patterns below for everything else.
Squares ending in 5
For a number n5 (like 35, 85, 115), multiply n by the next number, then write 25 after it.
Why: (10n + 5)2 = 100 n2 + 100 n + 25 = 100 n (n + 1) + 25.
- 8 × 9 = 72.n times the next number.
- Write 25 after: 7225.
- 11 × 12 = 132.n can have two digits.
- 13225.
Squares near 50 and near 100
Near 50: for 50 + k, the first part is 25 + k and the last two digits are k2. 532 = 28 | 09 = 2809. 472 = 22 | 09 = 2209.
Near 100: for 100 + k, the first part is 100 + 2k and the last two digits are k2. 962 = 92 | 16 = 9216. 1082 = 116 | 64 = 11664.
If k2 has more than two digits, carry: 1132 = 126 | 169 → 127 | 69 = 12769.
- k = 7. First part 25 + 7 = 32.Near 50 rule.
- Last two digits 72 = 49.
- 3249.
- k = -7. First part 100 - 14 = 86.Near 100 rule; 2k = -14.
- Last two digits 49.
- 8649.
Any 2-digit square: a2 | 2ab | b2
For a two-digit number ab, write three blocks: a2, then 2 × a × b, then b2, and carry from right to left, one digit per block (except the left one).
672: 36 | 84 | 49 → write 9 carry 4; 84 + 4 = 88, write 8 carry 8; 36 + 8 = 44. Answer 4489.
- Blocks: 49 | 42 | 9.a2 = 49, 2ab = 42, b2 = 9.
- 9 stays. 42: write 2, carry 4. 49 + 4 = 53.
- 5329.
Square roots of perfect squares
Step 1: the last digit of the square tells you two possible last digits of the root. 1 → 1 or 9, 4 → 2 or 8, 9 → 3 or 7, 6 → 4 or 6, 5 → 5, 0 → 0.
Step 2: cover the last two digits. The number left tells you the tens digit: find the biggest square below it.
Step 3: choose between the two candidates by comparing with the square of (tens digit)5.
| Square ends in | 1 | 4 | 5 | 6 | 9 | 0 |
|---|---|---|---|---|---|---|
| Root ends in | 1 or 9 | 2 or 8 | 5 | 4 or 6 | 3 or 7 | 0 |
- Ends in 4, so the root ends in 2 or 8.22 = 4 and 82 = 64 both end in 4.
- Cover 44: 77 remains. 82 = 64 ≤ 77 < 81, so the tens digit is 8.The root is in the 80s.
- 852 = 7225, which is less than 7744, so take the larger candidate: 88.Above the halfway square means the bigger ending.
Cube roots of perfect cubes
Last digits of cubes are all different, so the last digit of a cube gives the last digit of the root directly. Only 2 <→ 8 and 3 <→ 7 swap; every other digit stays the same.
Cover the last three digits. The number left tells you the first digit: the biggest cube below it.
| Cube ends in | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| Root ends in | 0 | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 |
- Ends in 6, so the root ends in 6.63 = 216 ends in 6.
- Cover 616: 175 remains. 53 = 125 ≤ 175 < 216, so the first digit is 5.
- 56.
Formula summary
Watch it step by step
Shortcuts and tricks
Square root of a 4-digit perfect square
Use it when: e.g. root of 7744.
- Last digit 4 → root ends 2 or 8.
- Front block 77 sits between 64 (82) and 81 (92) → tens digit 8.
- Choose between 82 and 88: 852 = 7225 < 7744, so the larger one.
- Ends 2/8, tens 8, above 852.
Cube root of a 6-digit perfect cube
Use it when: e.g. cube root of 175616.
- Last digit 6 → root ends 6.
- Strike the last three digits: 175. 53 = 125 ≤ 175 < 216 = 63, so tens digit 5.
- 56.
Before you move on, you should be able to...
- square any number ending in 5, near 50 or near 100 in your head
- find the square root of a 4-digit perfect square in 15 seconds
- find the cube root of a 6-digit perfect cube in 10 seconds
Practice questions (17)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy 952
- 9 × 10 | 25.
Answer: 9025
easy 1152
- 11 × 12 = 132 | 25.
Answer: 13,225
easy 562
- 25 + 6 = 31 | 36.
Answer: 3136
easy 442
- 25 - 6 = 19 | 36.
Answer: 1936
medium 932
- 100 - 14 = 86 | 49.
Answer: 8649
medium 1132
- 100 + 26 = 126 | 169 → 126 + 1 carry = 127 | 69.
Answer: 12,769
medium Square root of 9409
- Ends 9 → 3 or 7.
- 94 between 81 and 100 → 9.
- 952 = 9025 < 9409 → 97.
Answer: 97
medium Square root of 5329
- Ends 9 → 3/7.
- 53 between 49 and 64 → 7.
- 752 = 5625 > 5329 → 73.
Answer: 73
hard Cube root of 571787
- Ends 7 → 3.
- 571 between 512 (83) and 729 → 8.
- 83.
Answer: 83
hard Cube root of 438976
- Ends 6 → 6.
- 438 between 343 and 512 → 7.
- 76.
Answer: 76
hard 9982
- Base 1000, k = -2: 1000 - 4 = 996 | 004.
Answer: 9,96,004
hard Square root of 15129
- Ends 9 → 3 or 7.
- Strike the last two digits: 151 is between 144 (122) and 169 → 12.
- 1252 = 15625 > 15129, so the smaller: 123.
Answer: 123
hard Cube root of 912673
- Ends 3 → root ends 7.
- 912 is between 729 (93) and 1000 → 9.
- 97.
Answer: 97
hard Square root of 65536
- Ends 6 → 4 or 6.
- 655 between 625 (252) and 676 → 25.
- 2552 = 65025 < 65536 → larger: 256.
Answer: 256
hard 9972
- Base 1000, k = -3: 1000 - 6 = 994 | 009.
Answer: 9,94,009
hard 10052
- Ends in 5: 100 × 101 = 10100 | 25.
Answer: 10,10,025
hard 1062 + 942
- 1062 = 112|36 = 11236.
- 942 = 88|36 = 8836.
- Sum 20072.
- (Or: 2 × (1002 + 62) = 20072.)
Answer: 20,072
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Squares 11 to 30 round the room (cheat sheet). |
| 15 min | Ending in 5, with the algebra on the board. |
| 20 min | Near 50 and near 100, with carries. |
| 15 min | a2 | 2ab | b2 for any two-digit square. |
| 20 min | Square roots of 4-digit squares: three steps. |
| 15 min | Cube roots of 6-digit cubes. |
| 20 min | Practice set, timed. |
| 5 min | Recap. |