📊 Squares, Cubes & Root Tables
Memorise squares up to 50² and cubes up to 20³ to turn root questions into 5-second solves
Why Memorise Squares & Cubes?
Here is the single biggest time-saving secret for Simplification & Approximation:
If you know that 28² = 784, then when the exam asks √784 = ?, you write 28 in under 2 seconds. Zero calculation.
Root and power questions appear in every banking and government exam paper. Students who calculate them waste 30–60 seconds per question. Students who have memorised the tables solve them in 5 seconds flat — and move on.
The investment is small: spend a few days drilling these tables, and you gain a permanent speed advantage for every exam you ever take.
Squares Table (1² to 50²)
Read through this table once, then use the patterns in the next section to lock it into memory.
| n | n² | n | n² | n | n² | n | n² |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 14 | 196 | 27 | 729 | 40 | 1600 |
| 2 | 4 | 15 | 225 | 28 | 784 | 41 | 1681 |
| 3 | 9 | 16 | 256 | 29 | 841 | 42 | 1764 |
| 4 | 16 | 17 | 289 | 30 | 900 | 43 | 1849 |
| 5 | 25 | 18 | 324 | 31 | 961 | 44 | 1936 |
| 6 | 36 | 19 | 361 | 32 | 1024 | 45 | 2025 |
| 7 | 49 | 20 | 400 | 33 | 1089 | 46 | 2116 |
| 8 | 64 | 21 | 441 | 34 | 1156 | 47 | 2209 |
| 9 | 81 | 22 | 484 | 35 | 1225 | 48 | 2304 |
| 10 | 100 | 23 | 529 | 36 | 1296 | 49 | 2401 |
| 11 | 121 | 24 | 576 | 37 | 1369 | 50 | 2500 |
| 12 | 144 | 25 | 625 | 38 | 1444 | ||
| 13 | 169 | 26 | 676 | 39 | 1521 |
Patterns in Squares — Smart Memorisation
Don't try to brute-force memorise all 50 values. Use these patterns to derive any square you forget.
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Why Memorise Squares & Cubes?
Here is the single biggest time-saving secret for Simplification & Approximation:
If you know that 28² = 784, then when the exam asks √784 = ?, you write 28 in under 2 seconds. Zero calculation.
Root and power questions appear in every banking and government exam paper. Students who calculate them waste 30–60 seconds per question. Students who have memorised the tables solve them in 5 seconds flat — and move on.
The investment is small: spend a few days drilling these tables, and you gain a permanent speed advantage for every exam you ever take.
Squares Table (1² to 50²)
Read through this table once, then use the patterns in the next section to lock it into memory.
| n | n² | n | n² | n | n² | n | n² |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 14 | 196 | 27 | 729 | 40 | 1600 |
| 2 | 4 | 15 | 225 | 28 | 784 | 41 | 1681 |
| 3 | 9 | 16 | 256 | 29 | 841 | 42 | 1764 |
| 4 | 16 | 17 | 289 | 30 | 900 | 43 | 1849 |
| 5 | 25 | 18 | 324 | 31 | 961 | 44 | 1936 |
| 6 | 36 | 19 | 361 | 32 | 1024 | 45 | 2025 |
| 7 | 49 | 20 | 400 | 33 | 1089 | 46 | 2116 |
| 8 | 64 | 21 | 441 | 34 | 1156 | 47 | 2209 |
| 9 | 81 | 22 | 484 | 35 | 1225 | 48 | 2304 |
| 10 | 100 | 23 | 529 | 36 | 1296 | 49 | 2401 |
| 11 | 121 | 24 | 576 | 37 | 1369 | 50 | 2500 |
| 12 | 144 | 25 | 625 | 38 | 1444 | ||
| 13 | 169 | 26 | 676 | 39 | 1521 |
Patterns in Squares — Smart Memorisation
Don't try to brute-force memorise all 50 values. Use these patterns to derive any square you forget.
Pattern 1: Last Digit Cycle
The unit digit of perfect squares follows a fixed repeating cycle of 10:
1, 4, 9, 6, 5, 6, 9, 4, 1, 0
This means:
- Numbers ending in 1 or 9 → square ends in 1
- Numbers ending in 2 or 8 → square ends in 4
- Numbers ending in 3 or 7 → square ends in 9
- Numbers ending in 4 or 6 → square ends in 6
- Numbers ending in 5 → square ends in 5
- Numbers ending in 0 → square ends in 0
Exam use: If a question asks √7056, and the options are 82, 84, 86, 88 — the answer MUST end in a digit whose square gives 6. That's 4 or 6. So the answer is either 84 or 86. You've eliminated two options instantly.
Pattern 2: Numbers Ending in 5
This is the fastest trick in the entire chapter.
n5² = n × (n + 1), then append 25
| Example | Working | Answer |
|---|---|---|
| 15² | 1 × 2 = 2, append 25 | 225 |
| 25² | 2 × 3 = 6, append 25 | 625 |
| 35² | 3 × 4 = 12, append 25 | 1225 |
| 45² | 4 × 5 = 20, append 25 | 2025 |
| 65² | 6 × 7 = 42, append 25 | 4225 |
| 85² | 8 × 9 = 72, append 25 | 7225 |
| 95² | 9 × 10 = 90, append 25 | 9025 |
| 105² | 10 × 11 = 110, append 25 | 11025 |
This works for ANY number ending in 5, no matter how large.
Pattern 3: Numbers Near 50
(50 ± n)² = 2500 ± 100n + n²
More precisely:
- (50 + n)² = 2500 + 100n + n²
- (50 − n)² = 2500 − 100n + n²
| Example | Working | Answer |
|---|---|---|
| 52² | 2500 + 200 + 4 | 2704 |
| 53² | 2500 + 300 + 9 | 2809 |
| 48² | 2500 − 200 + 4 | 2304 |
| 47² | 2500 − 300 + 9 | 2209 |
| 56² | 2500 + 600 + 36 | 3136 |
| 44² | 2500 − 600 + 36 | 1936 |
This extends your effective range well beyond the table.
Pattern 4: Numbers Near 100
(100 + n)² = 10000 + 200n + n²
| Example | Working | Answer |
|---|---|---|
| 101² | 10000 + 200 + 1 | 10201 |
| 103² | 10000 + 600 + 9 | 10609 |
| 105² | 10000 + 1000 + 25 | 11025 |
| 98² | 10000 − 400 + 4 | 9604 |
| 97² | 10000 − 600 + 9 | 9409 |
Pattern 5: Difference of Consecutive Squares
n² − (n−1)² = 2n − 1
If you know 24² = 576, then:
- 25² = 576 + (2×25 − 1) = 576 + 49 = 625 ✓
This lets you hop from any known square to its neighbour.
Cubes Table (1³ to 20³)
Cubes appear less frequently than squares, but when they do, they're worth easy marks if you know the table.
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 11 | 1331 |
| 2 | 8 | 12 | 1728 |
| 3 | 27 | 13 | 2197 |
| 4 | 64 | 14 | 2744 |
| 5 | 125 | 15 | 3375 |
| 6 | 216 | 16 | 4096 |
| 7 | 343 | 17 | 4913 |
| 8 | 512 | 18 | 5832 |
| 9 | 729 | 19 | 6859 |
| 10 | 1000 | 20 | 8000 |
Key cubes to absolutely memorise: 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729. These appear the most in exams.
Cube Root Shortcuts
Method 1: Factor Out Thousands
³√(n × 1000) = ³√n × 10
- ³√8000 = ³√8 × 10 = 20
- ³√27000 = ³√27 × 10 = 30
- ³√125000 = ³√125 × 10 = 50
This works because 1000 = 10³.
Method 2: Last-Digit Method for Perfect Cubes
The last digit of a perfect cube tells you the last digit of the cube root — with a twist:
| Cube ends in | Cube root ends in |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 7 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 3 |
| 8 | 2 |
| 9 | 9 |
Memory trick: 0, 1, 4, 5, 6, 9 map to themselves. Only 2↔8 and 3↔7 swap (they are complements summing to 10).
Step-by-step for ³√17576:
- Last digit is 6 → cube root ends in 6
- Remove last three digits: 17 remains
- Find the largest cube ≤ 17: 2³ = 8, 3³ = 27 (too big) → so it's 2
- Answer: 26 ✓ (verify: 26³ = 17576)
Another example — ³√39304:
- Last digit is 4 → cube root ends in 4
- Remove last three digits: 39 remains
- Largest cube ≤ 39: 3³ = 27 ✓, 4³ = 64 (too big) → so it's 3
- Answer: 34 ✓ (verify: 34³ = 39304)
This method finds the cube root of any perfect cube (up to 99³) in seconds.
Square Root Estimation (Non-Perfect Squares)
In approximation questions, you often need √n where n is NOT a perfect square. Here's how:
Step-by-Step Method
- Find the nearest perfect square below and above n
- Use linear interpolation between them
√n ≈ √a + (n − a) / (2√a)
where a is the nearest perfect square ≤ n.
Example: √200
- Nearest perfect square: 196 = 14²
- √200 ≈ 14 + (200 − 196) / (2 × 14) = 14 + 4/28 = 14 + 0.14 ≈ 14.14 ✓
Example: √75
- Nearest perfect square: 81 = 9² (above), 64 = 8² (below)
- √75 ≈ 8 + (75 − 64) / (2 × 8) = 8 + 11/16 = 8 + 0.69 ≈ 8.66 ✓
Example: √500
- Nearest perfect square: 484 = 22²
- √500 ≈ 22 + (500 − 484) / (2 × 22) = 22 + 16/44 = 22 + 0.36 ≈ 22.36 ✓
For exam approximation, this level of accuracy (2 decimal places) is more than sufficient.
Simplification Trick
Sometimes you can simplify the root first:
- √200 = √(100 × 2) = 10√2 ≈ 10 × 1.414 = 14.14
- √75 = √(25 × 3) = 5√3 ≈ 5 × 1.732 = 8.66
- √72 = √(36 × 2) = 6√2 ≈ 6 × 1.414 = 8.49
Memorise these three values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236
Speed Tips for Exams
Tip 1: Reverse Lookup is the Real Skill
The exam rarely asks "What is 37²?" — it asks "What is √1369?" or embeds 1369 inside a larger expression. Train yourself to recognise squares on sight, not just calculate them forward.
Drill method: Write squares on flashcards with the square on the front and the root on the back. Quiz yourself in reverse.
Tip 2: Eliminate Options Using Last Digit
Before calculating, check which options are even possible:
- √2916 → ends in 6, so root ends in 4 or 6
- Options: 52, 54, 56, 58 → only 54 or 56 possible
- Quick check: 54² = 2916 ✓
Tip 3: Use Range Estimation
If you need √1850 approximately:
- 42² = 1764 and 43² = 1849 and 44² = 1936
- 1850 is just above 1849, so √1850 ≈ 43 (just over)
Tip 4: Break Complex Expressions
When you see √(a × b), check if either factor is a perfect square:
- √(4 × 196) = 2 × 14 = 28
- √(9 × 225) = 3 × 15 = 45
- ³√(8 × 343) = 2 × 7 = 14
Tip 5: Daily Practice Schedule
| Week | Focus | Minutes/Day |
|---|---|---|
| Week 1 | Squares 1–25 (forward and reverse) | 10 |
| Week 2 | Squares 26–50 + Cubes 1–10 | 10 |
| Week 3 | Cubes 11–20 + All patterns | 10 |
| Week 4 | Mixed reverse recognition drills | 10 |
After 4 weeks, you should be able to recognise any square (1–50) or cube (1–20) within 2 seconds.