Indices & Standard Form (Years 8–11) — Maths

Computer memory doubles and doubles (2, 4, 8, 16 gigabytes…), a virus multiplies, the distance to the Sun is 150 million km, an atom is 0.0000000001 m across. Writing all those zeros is painful, so mathematicians use powers. An index (plural indices), or power, is shorthand for multiplying a number by itself: instead of 2 × 2 × 2 × 2 × 2 we write 2⁵ — "2 to the power 5". The small raised number says how many times to multiply.

Why this matters: powers are how we tame very large and very small numbers, and the index laws are the exact rules you'll reuse all through algebra and science. Standard form is the everyday notation of astronomy, chemistry and computing.

2⁵ means multiply 2 by itself 5 times = 32

2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The 2 is the base; the 5 is the power (index).

The index laws (and why they work)

Multiplying just lines the factors up, so you simply add the powers: 2² × 2³ = 2⁵.

Because a power is just repeated multiplication, the laws are really just counting how many of the base you have. Line up the factors and the rule appears:

Multiplying: aᵐ × aⁿ = aᵐ⁺ⁿ — add the powers. (2² × 2³ = 2×2 × 2×2×2 = 2⁵.)
Dividing: aᵐ ÷ aⁿ = aᵐ⁻ⁿ — subtract the powers (the ones on top cancel the ones underneath).
Power of a power: (aᵐ)ⁿ = aᵐⁿ — multiply the powers.

Operation	Rule	Example
Multiply	add powers	x⁵ × x³ = x⁸
Divide	subtract powers	5⁶ ÷ 5⁴ = 5² = 25
Power of a power	multiply powers	(10²)³ = 10⁶

Simplify x⁵ × x³ ÷ x²: add then subtract the powers → 5 + 3 − 2 = 6, so the answer is x⁶.

Common mistake: the laws only work for the same base. You can combine powers of 2 with powers of 2, but 2³ × 3² must be worked out separately (8 × 9 = 72) — you cannot add those powers.

Two special powers

Each step down divides by 2, so past 2¹ you reach 2⁰ = 1, then 2⁻¹ = 1/2.

a⁰ = 1 — anything to the power 0 is 1. (It follows from dividing: aⁿ ÷ aⁿ = a⁰, but also equals 1.)
a⁻ⁿ = 1/aⁿ — a negative power means "one over". So 2⁻³ = 1/2³ = 1/8.

Standard form

The power counts the point's jumps: 68000 = 6.8 × 10⁴ (front number always between 1 and 10).

Scientists write very large or very small numbers compactly as A × 10ⁿ, where A is between 1 and 10. The power of 10 just records how far the decimal point has shifted.

Big number → positive power (how many places the point moves left). Small number → negative power (point moves right).

68 000 = 6.8 × 10⁴ (point moved 4 places). 0.00042 = 4.2 × 10⁻⁴ (4 places the other way). Notice the front number always sits between 1 and 10.

Memory hook: powers are repeated multiplication; same-base powers add when you multiply and subtract when you divide; standard form keeps the front number between 1 and 10 — positive power for big, negative for small.