Beyond Rankers Academy
Building Minds, Not Just Rankers
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Beyond Rankers Academy
Building Minds, Not Just Rankers
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Multiply a number by itself and you get a square. Do it once more and you get a cube. Both hide beautiful patterns — and knowing the first twenty by heart makes the rest of Class 8 far easier.
A perfect square is what you get by multiplying a whole number by itself: 1, 4, 9, 16, 25 …
🔲 Learn these by heart
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 8 | 64 | 15 | 225 |
| 2 | 4 | 9 | 81 | 16 | 256 |
| 3 | 9 | 10 | 100 | 17 | 289 |
| 4 | 16 | 11 | 121 | 18 | 324 |
| 5 | 25 | 12 | 144 | 19 | 361 |
| 6 | 36 | 13 | 169 | 20 | 400 |
| 7 | 49 | 14 | 196 | 25 | 625 |
🔍 Three patterns worth knowing
A square never ends in 2, 3, 7 or 8. If a number does, it is not a perfect square.
The gaps between squares are the odd numbers: 1, +3, 4, +5, 9, +7, 16 …
The sum of the first n odd numbers is n²: 1+3+5+7 = 16 = 4².
A one-second check: is 4358 a perfect square? It ends in 8, so no — you do not need to compute anything.
The square root asks the reverse question: which number, multiplied by itself, gives this?
√ Finding a root by prime factors
Break the number into prime factors: 1024 = 2×2×2×2×2×2×2×2×2×2.
Group them in pairs: (2×2)(2×2)(2×2)(2×2)(2×2).
Take one from each pair and multiply: 2×2×2×2×2 = 32. So √1024 = 32.
Every positive number has two square roots, one positive and one negative: 5² = 25 and (−5)² = 25. The symbol √ always means the positive one, so √25 = 5, not ±5.
A cube is a number multiplied by itself three times. Unlike squares, a cube can end in any digit.
🧊 The cubes to know
| n | n³ | n | n³ |
|---|---|---|---|
| 1 | 1 | 7 | 343 |
| 2 | 8 | 8 | 512 |
| 3 | 27 | 9 | 729 |
| 4 | 64 | 10 | 1000 |
| 5 | 125 | 11 | 1331 |
| 6 | 216 | 12 | 1728 |
Cube roots by grouping in threes: 3375 = 3×3×3 × 5×5×5. One from each group of three gives 3 × 5 = 15. So ∛3375 = 15.
The last digit gives the cube root away. A cube ending in 8 has a root ending in 2 (2³=8); ending in 7 → root ends in 3; ending in 2 → root ends in 8. Only 2, 3, 7 and 8 swap like this; 0, 1, 4, 5, 6 and 9 stay the same.
12 questions on squares, cubes and their roots. Work on paper, not in your head. 🍀