Edexcel IAL Pure 1
Laws of Indices & Surds
Master the language of powers and exact numbers — the toolkit every algebraic manipulation builds on.
3 Topics
Pure 1
Core Foundation
~45 min study
What's in this chapter
9 index laws — how to combine, split and flip powers
Changing the base to put terms on equal footing
What surds are and why we use them for exact answers
Simplifying surds using square factors
Collecting like surd terms
Rationalising denominators — all 3 cases
What is an index (power / exponent)?
In the expression an, the number a is the base and n is the index (also called the power or exponent). The index tells you how many times the base is multiplied by itself.
Index laws are rules that let you simplify expressions involving powers without working them all out. They are the backbone of algebra at every level beyond GCSE.
Think of it like this: If exponents are shorthand for repeated multiplication, then index laws are the grammar rules that let you combine that shorthand efficiently — like contracting "cannot" to "can't" in writing.
The 9 Index Laws
These apply whenever the base is the same. Learn them individually — then you'll spot which one to use automatically.
am × an = am+n
Multiply → add indices
am ÷ an = am−n
Divide → subtract indices
(am)n = amn
Power of a power → multiply
(ab)n = anbn
Power over a product
a1 = a
Power of 1
a0 = 1
Zero power (a ≠ 0)
a1/m = m√a
Fractional index → root
am/n = n√(am)
Root then power (or power then root)
a−m = 1am
Negative index → reciprocal
Critical rule: Index laws only work when terms share the same base. You cannot simplify 23 × 52 using index laws — different bases mean no shortcut.
Understanding the Tricky Ones
Fractional indices & roots: a1/2 = √a because when you square a1/2 you get a1/2 × 2 = a1 = a — exactly what squaring a square root does. So the fractional index is just a root in disguise.
For am/n: you can do it either order. 272/3 = (3√27)2 = 32 = 9. Root first, then power is usually easier because the numbers stay smaller.
Negative indices: a−m just means "flip it over" (take the reciprocal). So 4−2 = 1/16. A negative index never makes the number negative — it makes it a fraction.
Zero index: Any non-zero number raised to the power 0 equals 1. You can see why from the division law: a3 ÷ a3 = a0 and anything divided by itself is 1.
Changing the Base
Sometimes you'll get an expression like 94 ÷ 37. They look like different bases, so you can't use the division law... yet. The trick is to rewrite 9 as a power of 3 first:
The key step is recognising that 9 = 32, so you substitute this in and apply the power-of-a-power law. Similarly, 8 = 23, 25 = 52, 27 = 33, 32 = 25 — memorise these for speed.
Strategy:
Worked Example
127
3
19
−2
n