SolidNumber
Numbers & EstimationPercentage, Rate & RatioAlgebraic ManipulationEquations & InequalitiesExam & ChallengeTimed quiz
Home / Lessons / Algebraic Manipulation

Laws of integral indices

2026-10-09Algebraic ManipulationMr Ng

Master product, quotient, power and zero-index laws for Secondary 3 integral indices.

Key formulas

am×an=am+na^m × a^n = a^{m+n}
am÷an=am−na^m ÷ a^n = a^{m−n}
(am)n=amn(a^m)^n = a^{mn}
a0=1(a≠0)a^{0} = 1 (a \neq 0)
a−n=1/ana^{−n} = 1/a^n

Same base is required

Index laws apply when bases match. Rewrite numbers as powers of the same base when possible: 8 = 2³, 9 = 3², 100 = 10².

Negative and zero indices

a^0 = 1 for a ≠ 0. A negative index means reciprocal: a^{−n} = 1/a^n. Do not confuse the sign of the index with the sign of the base.

Power of a product

(ab)^n = a^n b^n and (a/b)^n = a^n / b^n. Apply the outer index to every factor inside.

Exam habit

Convert everything to prime bases before combining indices. Finish by writing positive indices unless the question allows negative indices.

Worked example

Same-base simplification

Problem. Simplify (3^4 × 3^{−1}) ÷ 3^2.

Solution. 3^{4+(−1)} ÷ 3^2 = 3^3 ÷ 3^2 = 3^{1} = 3.

Review quiz

1 / 4

3^5 × 3^2 =

Practice questions

  1. Simplify 3^5 × 3^2 and 3^5 ÷ 3^2.
  2. Evaluate 5^{−2} and 7^0.
  3. Simplify (2^3)^4 × 2^{−5}.

Study notes

Is (−2)^0 equal to −1?

No. Any non-zero number to the power 0 equals 1, so (−2)^0 = 1.

How do I handle 2^3 × 4^2?

Write 4 as 2² so 4^2 = (2²)^2 = 2^4, then add indices: 2^3 × 2^4 = 2^7.

More in Algebraic Manipulation