Laws of Indices, Every Rule With a Worked Example

Indices are one of the first places in Secondary math where a small rule, applied with confidence, saves a long calculation. They are also one of the first places where a half-remembered rule quietly produces a wrong answer. The fix is not to memorise harder; it is to see why each law is true, because a rule you understand is a rule you can rebuild when memory fails.

Algebraic manipulation, indices included, is the single most-tested topic across the E-Math papers GPA has counted, present in every year. It is worth getting watertight.

The three core laws

Multiplying powers of the same base, you add the indices:

am x an = am+n. For example 23 x 24 = 27, because three twos times four twos is seven twos.

Dividing, you subtract:

am / an = am-n.

A power of a power, you multiply:

(am)n = amn. For example (32)3 = 36, because 32 taken three times over multiplies the indices.

Each of these is just counting how many times the base appears, which is why they are worth seeing rather than memorising.

The three students get wrong

The zero power. Anything (except zero) to the power zero is 1: a0 = 1. The reason is the division law: am / am is both a0 and 1, so a0 must be 1. So 50 = 1, not 0.

Negative powers. A negative index means one over the positive index: a-n = 1/an. So 2-3 = 1/23 = 1/8. The minus sign moves the power to the denominator; it does not make the number negative.

Fractional powers. The denominator of the index is a root, and the numerator is a power: am/n = (nth root of a)m. So 272/3 is the cube root of 27, which is 3, squared, giving 9. And 163/4 is the fourth root of 16, which is 2, cubed, giving 8.

These three are where most index marks are lost, and all three follow from the same logic as the core laws, not from a separate act of memory.

A worked combination

Simplify 8-2/3. Take it apart in order: the cube root of 8 is 2, squaring gives 4, and the negative sign sends it to the denominator, so 8-2/3 = 1/4. Reading the index from the inside out, root then power then sign, keeps a compound case from turning into a guess.

The teaching point

Every index law is a way of counting how often a base appears, and the three that trip students up, the zero power, the negative power and the fractional power, all fall out of that same counting. A student who can rebuild the rule from the reasoning will not be the one who writes 50 = 0 under exam pressure. That is the difference GPA aims for: not more rules remembered, but rules understood well enough to reconstruct.

If your child applies the easy index laws but slips on negative or fractional powers, that is a clear, teachable gap. GPA's Secondary Math programme builds algebra from the reasoning up, on real O-Level papers, and a short diagnostic consult will show you exactly where the slips happen.

Every numerical example on this page was checked independently before publishing.