IB Maths AI HL Exponentials & Logs Paper 1 & 2 index laws ~6 min read

Laws of Indices

The laws of indices are the rules for working with powers — multiplying, dividing and raising them to further powers. They work on numbers and on algebra alike, and they are the foundation for everything in exponentials and logarithms.

📘 What you need to know

To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.
To divide, subtract the indices; to raise a power to a power, multiply them.
a⁰ = 1 and a¹ = a for any non-zero base.
A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ.
A fractional index means a root: a^1/n = ⁿ√a, and a^m/n = (ⁿ√a)^m.
The laws only work for the same base — rewrite bases as powers of a common base first.

The laws of indices

An index (or power) tells you how many times a base is multiplied by itself. The core laws come straight from that idea: a³ × a² writes out as five factors of a, so the indices simply add.

Multiplying powers of the same base just stacks the factors together — so the indices add. The other laws follow from the same counting idea.

Core index laws a^m × aⁿ = a^m+n · a^m ÷ aⁿ = a^m−n (a^m)ⁿ = a^mn · (ab)ⁿ = aⁿbⁿ · a⁰ = 1 these are not in the formula booklet — you must remember them

Negative and fractional powers

A negative index flips the term into its reciprocal. A fractional index is a root: the denominator gives the root, the numerator gives a power.

Negative and fractional powers a⁻ⁿ = 1aⁿ · a^1/n = ⁿ√a a^m/n = (ⁿ√a)^m = ⁿ√(a^m) denominator = the root — numerator = the power

For a fraction raised to a negative power, take the reciprocal of the fraction and use the positive power: (a/b)⁻ⁿ = (b/a)ⁿ.

Changing the base

The index laws only apply when terms share the same base — 2³ × 5² cannot be combined. Often you can rewrite one base as a power of another: since 4 = 2², a term in base 4 can be turned into a term in base 2.

Changing the base: e.g. 2⁵ × 4³ = 2⁵ × (2²)³ = 2⁵ × 2⁶ = 2¹¹. Once the bases match, the laws take over.

🧭 Recipe — simplifying with index laws

Match the bases — if they differ, rewrite one base as a power of another.
Clear the brackets: raise every factor inside a bracket to the outside power.
Combine same-base terms: add indices when multiplying, subtract when dividing.
Rewrite negative powers as reciprocals and fractional powers as roots.
Present the answer in the form requested — a single power, a fraction, or a number.

Worked examples

WE 1

Multiplying and dividing powers

Evaluate (3⁴ × 3²) ÷ 3³.

multiply: add the indices 3⁴ × 3² = 3⁴⁺² = 3⁶ divide: subtract the indices 3⁶ ÷ 3³ = 3⁶⁻³ = 3³ = 27 work with the indices first, then evaluate the single power at the end.

WE 2

A fraction to a negative power

Evaluate 32⁻⁴.

negative power ⇒ reciprocal of the fraction (³⁄₂)⁻⁴ = (²⁄₃)⁴ raise top and bottom to the power 4 = 2⁴ ÷ 3⁴ = 16 ÷ 81 1681 flip the fraction, then the negative power becomes positive.

WE 3

Fractional powers

Evaluate (a) 81^3/4 and (b) 64^−2/3.

(a) denominator 4 = 4th root, numerator 3 = power 81^3/4 = (⁴√81)³ = 3³ = 27 (b) negative ⇒ reciprocal; 3 = cube root 64^−2/3 = 1 ÷ (³√64)² = 1 ÷ 4² (a) 27 · (b) 116 take the root first (smaller numbers), then apply the power.

WE 4

Simplifying an algebraic fraction

Simplify (5a³b)(4a²b⁴)10a⁴b².

expand the numerator — add indices (5a³b)(4a²b⁴) = 20a⁵b⁵ divide by 10a⁴b² — subtract indices 20 ÷ 10 = 2, a⁵⁻⁴ = a, b⁵⁻² = b³ 2ab³ handle the numbers and each letter as separate calculations.

WE 5

Brackets and negative powers

Simplify (2x³y⁻²)⁴(4x⁵y⁻³)⁻¹, giving your answer with positive indices.

clear each bracket — multiply the indices (2x³y⁻²)⁴ = 16x¹²y⁻⁸ (4x⁵y⁻³)⁻¹ = ¹⁄₄x⁻⁵y³ multiply — add indices, with 16 × ¹⁄₄ = 4 4x¹²⁻⁵y⁻⁸⁺³ = 4x⁷y⁻⁵ 4x⁷y⁵ a negative index on a bracket flips it — then write y⁻⁵ as 1 ÷ y⁵.

WE 6

Full question: changing the base

(a) Write 8 and 16 as powers of 2. (b) Hence simplify (8² × 16) ÷ 4³, giving your answer as a power of 2.

(a) express each number in base 2 8 = 2³, 16 = 2⁴ (and 4 = 2²) (b) rewrite the expression in base 2 ((2³)² × 2⁴) ÷ (2²)³ = (2⁶ × 2⁴) ÷ 2⁶ = 2¹⁰ ÷ 2⁶ (a) 8 = 2³, 16 = 2⁴ · (b) 2⁴ change every base to 2 first — then the index laws do the rest.

💡 Top tips

Multiply ⇒ add indices, divide ⇒ subtract, power of a power ⇒ multiply.
A negative index never makes a number negative — it gives the reciprocal.
For a^m/n, take the root first — the numbers stay small and easy.
The index laws need the same base — rewrite bases as powers of a common base.
A power outside a bracket applies to every factor inside, the number included.

⚠ Common mistakes

Multiplying the indices when multiplying terms — a³ × a² = a⁵, not a⁶.
Forgetting the coefficient in a bracket — (2x³)⁴ = 16x¹², not 2x¹².
Treating a negative index as a negative number — 4⁻³ = ¹⁄₆₄, not −64.
Swapping the root and power in a^m/n — the denominator is the root.
Combining different bases — 2³ × 5² cannot be added; change the base first if possible.

Next up: Introduction to Logarithms — the inverse of a power. Indices and logarithms are two views of the same relationship: a^x = b is exactly x = log_ab. Master the index laws and the log laws will feel familiar.

Need help with AI HL Exponentials & Logs?

Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.

Book Free Session →