Multiply a number by itself and you get its square. Multiply it by itself twice more and you get its cube. Roots simply undo each one. See the patterns and use them on real problems.
Squares and cubes are among the simplest ideas in mathematics to define, yet they open the door to area, volume, the Pythagorean theorem and eventually algebra itself. The words are not an accident: a squared number really is the area of a square, and a cubed number really is the volume of a cube. This guide covers both directions in full, building squares and cubes, and undoing them with roots, with patterns, shortcuts and real problems along the way.
A square number (or perfect square) is what you get by multiplying a whole number by itself. It is 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. A square with sides 5 cm long has an area of 25 cm², which is where the name comes from.
| 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 |
Shortcuts for squaring bigger numbers:
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. The first n odd numbers always add up to 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 be a perfect square. No calculation is needed.
A third, surprising pattern: perfect squares are the only numbers with an odd number of factors. For example, 36 has nine factors (1, 2, 3, 4, 6, 9, 12, 18, 36), because the factor 6 pairs with itself.
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 is 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 is 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. Group the primes in pairs: (2 × 2), (2 × 2), (3 × 3). Take one from each pair: 2 × 2 × 3 = 12. Check: 12 × 12 = 144. ✓
(a) √400: 400 = 24 × 52 = (22)2 × 52, so √400 = 22 × 5 = 20.
(b) √1,764: 1,764 = 22 × 32 × 72, so √1,764 = 2 × 3 × 7 = 42. Check: 42 × 42 = 1,764. ✓
A quick no-factorising method for perfect squares: look at the last digit and bracket the size. For √1,296: 30² = 900 and 40² = 1,600, so the root is between 30 and 40. The last digit 6 means the root ends in 4 or 6. Try 36: 36² = 1,296. ✓
Most numbers are not 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. Test 7.1: 7.1² = 50.41, which is too big. Test 7.07: 7.07² = 49.98, which is very close. So √50 ≈ 7.07.
If a number is not a perfect square, you can still pull out any square factor. This keeps the root exact instead of a rounded decimal.
(a) √50 = √(25 × 2) = √25 × √2 = 5√2. (Check: 5 × 1.414 ≈ 7.07 ✓.)
(b) √72 = √(36 × 2) = 6√2. And √12 = √(4 × 3) = 2√3.
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, so three copies of the same number are multiplied together. It is 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. Note that 4³ is not 4 × 3 = 12.
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1,000 |
A curious pattern: the sum of the first n cubes is the square of the sum of the first n numbers. For n = 4: 1 + 8 + 27 + 64 = 100, and (1 + 2 + 3 + 4)² = 10² = 100. Also, the number 64 is both a square (8²) and a cube (4³), because 64 = 26.
A cube root undoes cubing. The cube root of 64 is the number that, used as a factor three times, 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 2s and one triple of 3s give 2 × 3 = 6. Check: 6 × 6 × 6 = 216. ✓
∛1,728: 1,728 = 26 × 33 = (22)3 × 33, so ∛1,728 = 22 × 3 = 12. Check: 12 × 12 × 12 = 1,728. ✓
Here cube roots behave 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, because a negative number cubed stays negative.
∛(−8) = −2, because (−2) × (−2) × (−2) = −8. Compare that to √(−9), which has no real answer at all.
This applies a square-root rule to cube roots, where it does not 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 5 cm side holds a volume of 5³ = 125 cubic centimetres. Squaring and cubing are not abstract. They are literally how area and volume are calculated.
A square field has an area of 2,025 m². How long is each side, and how much fence goes round it?
Side = √2,025 = 45 m (45² = 2,025). Perimeter = 4 × 45 = 180 m.
A cubic water tank holds 27,000 litres. How long is each edge? Since 1 m³ = 1,000 L, the volume is 27 m³. Edge = ∛27 = 3 m.
A rectangular field is 80 m long and 60 m wide. A path along the edges is 80 + 60 = 140 m. A diagonal path is the longest side of a right-angled triangle, so by the Pythagorean theorem its length is √(80² + 60²) = √(6,400 + 3,600) = √10,000 = 100 m. The shortcut saves 40 m.
If you double the side of a square from 2 to 4, the area goes from 4 to 16, which is four times as big. If you double the side of a cube from 2 to 4, the volume goes from 8 to 64, which is eight times as big. This is why a pizza twice as wide gives four times as much to eat, and why a cube twice as long needs eight times as much material to fill it.
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.
Roots can be split over multiplication, but not over addition.
The fix: add first, then take the root: √(9 + 16) = √25 = 5. The answer 7 is wrong, because 7² = 49, not 25. The splitting rule only works for products: √(9 × 16) = √144 = 12 = √9 × √16 = 3 × 4. ✓
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1. 13² = 169. √225 = 15 (15² = 225). ∛(−125) = −5, since (−5)³ = −125.
2. No. It ends in 8, and a perfect square can only end in 0, 1, 4, 5, 6 or 9.
3. √48 = √(16 × 3) = 4√3.
1. √169 = 13 and ∛343 = 7.
2. 1,764 = 22 × 32 × 72, so √1,764 = 2 × 3 × 7 = 42.
3. Side = √2,025 = 45 m. Perimeter = 4 × 45 = 180 m.
4. Diagonal = √(80² + 60²) = √10,000 = 100 m. Two sides = 140 m. The diagonal saves 140 − 100 = 40 m.
Use real squares and cubes: tiles, blocks and dice. Let learners build 4 × 4 squares from counters and add the L-shaped layers to discover the odd-number pattern for themselves. Build a 3 × 3 × 3 cube from small blocks and count the layers. Ask “does the answer make sense?” after every root, by squaring or cubing it back. Common errors are doubling instead of squaring, splitting roots over addition, and forgetting that √ means the positive root.
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 are 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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