Powers and Roots
Squares are literally squares and cubes are literally cubes. Start from the shapes, and the index laws, zero powers and negative powers stop being rules to memorise and become patterns you can see.
National curriculum KS3: square and cube numbers, square and cube roots, integer powers, index laws, standard form prerequisite1 · Squares are squares
5² is said "five squared" and means 5 × 5 = 25. The name is not a metaphor. If you build a square with 5 dots along each side, it contains exactly 25 dots. Drag the slider and count.
The square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Know them to 15². Notice what the orange L-shape shows: to go from one square to the next you add the next odd number (1, 3, 5, 7, …). That's why the differences between square numbers are always odd.
2 · Cubes are cubes
4³ is "four cubed" and means 4 × 4 × 4 = 64. Build a cube with 4 blocks along each edge: 4 layers, each a 4 × 4 square of 16, total 64.
The cube numbers to know: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. The dimensions are the clue to the names: a square is 2-dimensional so squaring uses the number twice; a cube is 3-dimensional so cubing uses it three times. After that the shapes run out but the pattern doesn't: n⁴ is n used four times, and so on.
3 · Reading the notation
The big number is the base: what you're multiplying. The small raised number is the index (plural indices; also called the power or exponent): how many copies to multiply. In Spanish, base and exponente, and the whole thing is a potencia.
The most common mistake in the whole topic: 2⁵ is not 2 × 5 = 10. It is five 2s multiplied, which is 32. Index means "how many copies", never "multiply by".
Negative bases. (−3)² = (−3) × (−3) = +9, because two negatives multiply to a positive. (−3)³ = −27: three negatives give a negative. Even powers of a negative are positive; odd powers stay negative. And watch the brackets: −3² means −(3²) = −9, but (−3)² = 9. Try base −3 above.
4 · Roots: going backwards
Squaring asks "5 along each side, how many in total?" The square root asks the reverse: "25 in total, how long is each side?" √25 = 5. The symbol √ is called the radical (raíz in Spanish). Likewise the cube root: ∛64 = 4, because a cube of 64 blocks has edge 4.
Two things to notice:
- Most numbers are not perfect squares, so their square roots are not whole numbers. √20 is between √16 = 4 and √25 = 5, so it's "4 point something" (4.47…). Trapping a root between two known squares is a skill examiners test.
- Every positive number has two square roots, because (−5)² is also 25. The √ symbol means the positive one. When an equation says x² = 25, the full answer is x = ±5.
5 · The index laws (don't memorise, count)
Each law is just counting copies. Change the indices and watch the copies line up.
| Law | Rule | Why |
|---|---|---|
| Multiplying | aᵐ × aⁿ = aᵐ⁺ⁿ | m copies then n more copies: m + n copies in total. |
| Dividing | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | n of the copies on the bottom cancel n on the top. |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | n groups, each holding m copies: m × n copies. |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 3)² = 2 × 3 × 2 × 3 = 2² × 3². |
The laws only work when the bases match. 2³ × 5² has no shortcut; you just work it out (8 × 25 = 200). And they are about multiplying and dividing: there is no law for 2³ + 2², which is simply 8 + 4 = 12.
6 · Zero and negative powers
What could 2⁰ or 2⁻¹ possibly mean? You can't multiply "zero copies" or "minus one copy". The answer comes from a pattern. Going down the staircase, each step divides by the base. Just keep going.
Reading the staircase:
- a⁰ = 1 for any base (except 0 itself). Not 0. Think of it as "no multiplying has happened yet", and the starting point for multiplication is 1.
- a⁻ⁿ = 1 ⁄ aⁿ. A negative index means reciprocal, not a negative number. 2⁻³ = ⅛, which is positive. 10⁻² = 0.01.
These make the dividing law hold everywhere. 2³ ÷ 2³ = 2⁰, and 8 ÷ 8 = 1, so 2⁰ must be 1. And 2² ÷ 2⁵ = 2⁻³, while 4 ÷ 32 = ⅛, so 2⁻³ must be ⅛. The pattern was forced on us; nobody chose it.
7 · Where this shows up
Area and volume
Area is measured in cm² and volume in cm³ precisely because they come from squaring and cubing lengths. Double a cube's edge and its volume goes up by 2³ = 8 times. That's why big animals need thick legs.
Standard form
The Sun's mass is 2 × 10³⁰ kg; a hydrogen atom is 1.7 × 10⁻²⁷ kg. Powers of ten, including negative ones, are how science writes very large and very small numbers. Next topic after this one.
Computing: binary and bytes
Each bit doubles the choices, so n bits give 2ⁿ patterns: 8 bits → 2⁸ = 256 values. 1 KiB = 2¹⁰ bytes, 1 MiB = 2²⁰, 1 GiB = 2³⁰. Colour depth, IP addresses and password strength are all powers of 2.
Compound interest
£1000 at 5% a year for 10 years is 1000 × 1.05¹⁰ ≈ £1629. Each year multiplies by the same factor, so years become an index.
Pythagoras
a² + b² = c² is squares and a square root in one line. Finding the hypotenuse always ends with √: a 3-4-5 triangle has c = √(9 + 16) = √25 = 5.
Earthquakes and sound
The Richter and decibel scales are powers of 10. A magnitude 7 quake shakes 10 times harder than a 6 and 100 = 10² times harder than a 5.
Worked examples
Simplify 3⁴ × 3² ÷ 3³
Estimate √40 without a calculator
Write 5⁻² and (2³)² as ordinary numbers
Exam-style: 2ˣ = 64. Find x.
Quick check
What you've learned
A power says how many copies of the base to multiply, and squares and cubes are named for the shapes they count. Roots run the process backwards. The index laws fall out of counting copies: add indices to multiply, subtract to divide, multiply for a power of a power, always with matching bases. Walking the staircase downwards forces a⁰ = 1 and a⁻ⁿ = 1/aⁿ. From here you are ready for standard form, surds, and the fractional indices that tie roots and powers together at GCSE.