Revision notes
Multiply, divide and raise powers quickly with the laws of indices — including zero, negative and fractional indices.
Ngozi forwards a message to 3 friends, and each of them forwards it to 3 more. After 8 rounds, 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 people get it. Writing 38 is much faster — and the laws of indices let us work with numbers like this in seconds.
In 25, the base is 2 and the index (or power) is 5. It means 2 × 2 × 2 × 2 × 2 = 32.
Multiplying powers of the same base: 23 × 24 is three 2s times four more 2s — seven 2s. So we add the indices: 23 + 4 = 27.
Dividing: 56 ÷ 52 cancels two of the six 5s, leaving four. So we subtract: 56 − 2 = 54.
Power of a power: (x2)3 = x2 × x2 × x2 = x6. So we multiply the indices.
🔑 Key idea: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ — the base must be the same.
73 ÷ 73 = 70 by the division law — but any number divided by itself is 1. So a0 = 1 (for any a that is not 0).
20 ÷ 23 = 2−3, and that is 1 ÷ 23 = 18. A negative index means 'one over' — it does NOT make the answer negative.
For a fraction, the negative index flips it: (23)−2 = (32)2 = 94.
🔑 Key idea: a0 = 1 and a−n = 1 ÷ aⁿ.
In 82/3, the bottom number 3 means cube root and the top number 2 means square. The cube root of 8 is 2, and 22 = 4. Taking the root first keeps the numbers small.
To solve 2x = 32, write 32 as a power of 2: 32 = 25. When the bases are equal, the indices must be equal, so x = 5.
🔑 Key idea: am/n: take the nth root, then raise to the power m.
Evaluate 28 × 21.
Step 1: Add the indices
28 × 21 = 28 + 1 = 29
Why? 8 factors of 2 times 1 more factors of 2 makes 9 factors.
Step 2: Work out the power
29 = 512
Answer: 512
Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.