Algebraic Roots & Indices

Summary

• Index laws are shortcut rules for multiplying, dividing, and raising powers to powers — they work identically for numbers and algebra.
• Anything to the power 1 is itself; anything to the power 0 is 1.
• Negative powers mean "flip it" — a⁻ⁿ = 1/aⁿ.
• Fractional powers mean "root it" — a^(1/n) = ⁿ√a, and more generally a^(m/n) = (ⁿ√a)ᵐ.
• You can combine negative and fractional powers together — negative flips it, the denominator of the fraction tells you the root, the numerator tells you the power.
• To find an unknown inside a power, get both sides of the equation over the same base, then set the powers equal to each other.

1. The Laws of Indices

Think of an index (or "power," or "exponent") as shorthand for repeated multiplication. c³ just means "multiply c by itself 3 times": c × c × c. Every index law below is really just a shortcut that falls straight out of that idea — if you ever forget a rule mid-exam, you can rebuild it by writing the powers out longhand.

The power of 1 and the power of 0

This one trips people up because it feels like it should be 0. But think about the division rule below and it'll make sense: a³ ÷ a³ = a⁰, and anything divided by itself is 1. So a⁰ must equal 1.

Multiplying and dividing powers with the same base

Raising a power to another power

This is different from the multiplication rule above — don't mix them up! a³ × a² = a⁵ (add), but (a³)² = a⁶ (multiply). The bracket is the giveaway: brackets around a power mean "multiply the powers."

Powers of products and fractions

2. Negative Indices — "Flip It"

A negative power doesn't mean the answer is negative — it means reciprocal (flip the fraction upside down). It's a natural extension of the division rule: if a² ÷ a⁵ = a⁻³, but longhand division tells you a²÷a⁵ = 1/a³, then a⁻³ and 1/a³ must be the same thing.

What to Memorise