Multiply a number by itself and you get its square. Multiply it by itself twice more and you get its cube. Square roots and cube roots simply undo each one.
Squares and cubes are two of the simplest ideas in math to define, yet they open the door to area, volume, the Pythagorean theorem, and eventually algebra itself. This guide covers both directions in full: building squares and cubes, and undoing them with roots.
A square number (or perfect square) is what you get by multiplying a number by itself. It's written with a small "2" above and to the right: 5² means 5 × 5.
5² = 5 × 5 = 25. The number 25 is a perfect square, and 5 is called its square root.
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Pattern 1 — Consecutive odd numbers: adding up odd numbers in order, starting from 1, always produces a perfect square.
1 = 1². 1+3 = 4 = 2². 1+3+5 = 9 = 3². 1+3+5+7 = 16 = 4². Adding the first n odd numbers always equals n².
Pattern 2 — Last digits: a perfect square can only ever end in 0, 1, 4, 5, 6, or 9. It can never end in 2, 3, 7, or 8.
This makes an instant test for ruling out perfect squares: 1,347 ends in 7, so it cannot possibly be a perfect square — no calculation needed.
A square root undoes squaring. The square root of 25 is the number that, multiplied by itself, gives 25 — which is 5. Square roots use the radical symbol: √25 = 5.
It's true that both 3 and −3 square to 9 — but the radical symbol has a specific, agreed meaning.
The fix: by convention, the radical symbol √ always refers to the positive square root, called the principal root. √9 = 3, not −3. If both roots are needed, it's written explicitly as ±√9 = ±3.
Break the number into prime factors, then pair them up. Each matching pair contributes one copy of that prime to the square root.
144 = 2×2×2×2×3×3 = (2×2) × (2×2) × (3×3). Pairing up: one 2, one 2, one 3 — wait, more precisely: pairs are (2,2), (2,2), (3,3), giving 2×2×3 = 12. Check: 12×12 = 144. ✓
Most numbers aren't perfect squares, and their square roots are irrational — a concept covered fully in our Number Systems article. Even without a calculator, you can estimate one by bracketing it between two perfect squares.
7² = 49 and 8² = 64. Since 50 sits just above 49, √50 is a little more than 7 — specifically about 7.07.
Tap any number below to see its square, cube, and whether it's a perfect square or perfect cube.
Tap a number above to see its full breakdown.
A cube number (or perfect cube) is what you get by multiplying a number by itself twice more — three copies of the same number multiplied together. It's written with a small "3": 4³ means 4 × 4 × 4.
4³ = 4 × 4 × 4 = 64. The number 64 is a perfect cube, and 4 is called its cube root.
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
A cube root undoes cubing. The cube root of 64 is the number that, multiplied by itself twice more, gives 64 — which is 4. Cube roots use a radical symbol with a small 3: ∛64 = 4.
Break the number into prime factors, then group them into triples instead of pairs. Each matching triple contributes one copy of that prime to the cube root.
216 = 2×2×2×3×3×3 = (2×2×2) × (3×3×3). One triple of 2's and one triple of 3's: 2×3 = 6. Check: 6×6×6 = 216. ✓
This is where cube roots behave completely differently from square roots. A negative number has no real square root, because no real number squared gives a negative result. But a negative number does have a real cube root — a negative number cubed stays negative.
∛(−8) = −2, because (−2)×(−2)×(−2) = −8. Compare that to √(−9), which has no real answer at all, since no real number squared can ever produce a negative result.
This applies a square-root rule to cube roots, where it doesn't hold.
The fix: squaring a negative always gives a positive (so negatives never have real square roots), but cubing a negative stays negative (so every negative number has a perfectly real cube root). ∛(−27) = −3 is a completely valid, real answer.
A square garden with a 12-metre side has an area of 12² = 144 square metres. A cube-shaped storage box with a 5cm side holds a volume of 5³ = 125 cubic centimetres. Squaring and cubing aren't abstract — they're literally how area and volume are calculated.
The word "square" for n² and "cube" for n³ come directly from geometry — a number squared gives the area of an actual square with that side length, and a number cubed gives the volume of an actual cube.
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Squares and square roots are the backbone of the Pythagorean theorem, distance formulas, and quadratic equations later in algebra. Cubes and cube roots show up anywhere volume matters — packaging, construction, and 3D design. Together, they're also the first real introduction to the idea of an "inverse operation" undoing another, a pattern that repeats throughout all of higher mathematics.
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