Laws of Indices

Multiplying powers of the same base

2³ × 2² means (2 × 2 × 2) × (2 × 2). Take the brackets away and there is one long row of 2s multiplied together: 3 copies beside 2 more copies, 5 copies in all. So 2³ × 2² = 2⁵.

Check with the values: 2³ = 8 and 2² = 4, and 8 × 4 = 32, which is 2⁵. When powers of the same base are multiplied, the copies join into one row, so the indices add. In general, aᵐ × aⁿ = aᵐ⁺ⁿ.

3 copies of 2 beside 2 more copies make 5 copies in all.

Dividing powers of the same base

Write 2⁵ / 2² as a fraction, with five 2s multiplied in the numerator and two 2s multiplied in the denominator. Each 2 in the denominator cancels one 2 in the numerator, because 2 ÷ 2 = 1. Two copies cancel, and 5 − 2 = 3 copies are left: 2⁵ / 2² = 2³.

Check with the values: 32 ÷ 4 = 8, which is 2³. When powers of the same base are divided, the indices subtract. In general, aᵐ / aⁿ = aᵐ⁻ⁿ.

Two of the five copies cancel with the two in the denominator, and 3 copies are left.

Larger indices

The copies do not need to be drawn once the laws are clear. 5² × 5⁴ = 5⁶, because 2 copies and 4 copies make 6. 7⁶ / 7² = 7⁴, because 2 of the 6 copies cancel.

A number on its own is one copy: 3 is 3¹. So 3² × 3 = 3³, and indeed 9 × 3 = 27, which is 3 × 3 × 3.

2 copies of 5 beside 4 more copies make 6 copies of 5.

The bases have to match

The laws work only when the bases are the same. 2³ × 3² is three 2s and two 3s multiplied: no 2 is a 3, so the copies cannot join into one power. Work out each power and multiply: 2³ = 8 and 3² = 9, so 2³ × 3² = 8 × 9 = 72.

The laws are about multiplying and dividing, not adding. 2³ + 2³ is 8 + 8 = 16, which is 2⁴, not 2⁶.

Three copies of 2 make 8.

Two copies of 3 make 9. They cannot join the row of 2s, so 2³ × 3² is only 8 × 9 = 72.

The usual mistakes

Multiplying the indices. 2³ × 2² is 2⁵, not 2⁶: the copies line up side by side, so the indices add.

Multiplying the bases. 2³ × 2² is not 4⁵. The base stays 2; only the number of copies changes.

Dividing the indices. 2⁶ / 2² is 2⁴, not 2³: dividing cancels copies one for one, so the indices subtract.

Worked example: Computer Storage Counted in Powers of 2

Question Computer storage is counted in powers of 2: 1 megabyte (MB) is 2²⁰ bytes and 1 gigabyte (GB) is 2³⁰ bytes. A memory card holds 2³⁵ bytes and one photograph takes up 2²² bytes. (a) How many photographs fit on the card? (b) A second card has twice the capacity of the first. Write its capacity in bytes as a power of 2, and find its capacity in gigabytes.

1.The number of photographs is the capacity of the card divided by the size of one photograph: 2³⁵ ÷ 2²².

The number of photographs is the capacity of the card divided by the size of one photograph.
2.Both powers have the base 2, so subtract the indices: 2³⁵ ÷ 2²² = 2^{35 − 22} = 2¹³.

The bases are the same, so subtract the indices: 2³⁵ ÷ 2²² = 2^{35 − 22} = 2¹³.
3.(a) 2¹³ = 2¹⁰ × 2³ = 1024 × 8 = 8192, so 8192 photographs fit on the card.

(a) 2¹³ = 1024 × 8 = 8192 photographs fit on the card.
4.The second card holds 2 × 2³⁵ bytes. The number 2 on its own is 2¹, and multiplying powers of the same base adds the indices: 2¹ × 2³⁵ = 2^{1 + 35} = 2³⁶ bytes.

The number 2 is 2¹, and multiplying adds the indices: 2¹ × 2³⁵ = 2³⁶ bytes.
5.(b) One gigabyte is 2³⁰ bytes, so the second card holds 2³⁶ ÷ 2³⁰ = 2^{36 − 30} = 2⁶ = 64 GB. Check: the first card holds 2³⁵ ÷ 2³⁰ = 2⁵ = 32 GB, and twice 32 is 64.

(b) 2³⁶ ÷ 2³⁰ = 2⁶ = 64 GB.

Answer: (a) 2¹³ = 8192 photographs; (b) 2³⁶ bytes, which is 64 GB

Common mistakes

Dividing the indices, 35 ÷ 22, or dividing the bases to get 1¹³. When powers of the same base are divided, the base stays and the indices are subtracted: 2^{35 − 22} = 2¹³.
Writing 2 × 2³⁵ as 4³⁵ or as 2⁷⁰. Doubling adds just one more factor of 2, so the index goes up by 1: 2 × 2³⁵ = 2³⁶.

More powers and roots problems, worked step by step →