Indices and Powers

La méthode, étape par étape

1. 1

   Multiplying the same base adds the indices

   2³ × 2⁴ = 2⁷

   Three twos multiplied by four twos is seven twos in total. The rule is a description of counting factors, not an arbitrary law, and a student who can say that sentence never confuses it with the multiply rule.

2. 2

   Dividing the same base subtracts the indices

   2⁷ ÷ 2⁴ = 2³

   Four of the seven twos cancel against the four on the bottom, leaving three. Division removes factors, which is why it subtracts — the reverse of the multiplication rule, for the reverse operation.

3. 3

   A power of a power multiplies the indices

   (2³)⁴ = 2¹²

   Four copies of 2³ multiplied together is four lots of three twos. This is the rule most often confused with the first one; the distinction is whether the powers are being multiplied together or one is being applied to the other.

4. 4

   Anything to the power zero is 1

   2³ ÷ 2³ = 2⁰, and also = 1, so 2⁰ = 1

   This is not a convention invented for convenience. Dividing any number by itself gives 1, and the subtraction rule says the answer is 2⁰ — so the two must be equal. Showing the derivation stops students misremembering it as 0.

5. 5

   A negative index means one over

   2⁻³ = 1/2³ = 1/8

   Continuing the division rule past zero forces this: 2² ÷ 2⁵ is 2⁻³, and cancelling directly gives 1/8. Negative indices make small numbers, not negative ones — a distinction that catches out a great many students.

Three laws, and everything else follows

There are really only three rules — add when multiplying, subtract when dividing, multiply when raising a power to a power — and the rest of the topic is consequences. Zero indices, negative indices and fractional indices all fall out of applying those three consistently past the point where the original definition made obvious sense.

That is worth saying explicitly, because the topic is often taught as a list of six or seven separate facts. Students who see three rules plus consequences have far less to remember and a way to reconstruct anything they forget.

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1
• a⁻ⁿ = 1/aⁿ
• a^(1/n) = the nth root of a

The rules only work on matching bases

2³ × 3⁴ cannot be simplified by adding indices, because the factors are not the same thing. The rules describe counting identical factors, and two threes and three twos have nothing to cancel or combine.

This is why rewriting to a common base is such a useful move: 8 × 4 becomes 2³ × 2² = 2⁵, and suddenly the rules apply. Exam questions in this topic are very often really testing whether a student spots that 8, 4 and 16 are all powers of 2.

Fractional indices are roots

A power of ½ is a square root, because x^½ × x^½ = x¹ by the addition rule — so x^½ is the thing that gives x when multiplied by itself, which is the definition of a square root. A power of ⅓ is a cube root by the same argument.

For a fraction like ⅔, take the root first and then the power: 8^(2/3) is the cube root of 8, which is 2, squared to give 4. Doing it in that order keeps the numbers small, which matters a great deal on a non-calculator paper.

How we teach indices

We derive the zero and negative index rules in front of the student rather than stating them. It takes four minutes and it converts two of the most-forgotten facts in GCSE maths into things that can be worked out again from the division rule.

We also drill recognising powers of 2, 3 and 5 on sight. A surprising number of index questions are straightforward once a student notices that 64 is 2⁶, and completely opaque otherwise.