Spotting squares and cubes

Knowing that 7^² = 49 lets us recognise 49 as a square number. We're reading the same fact backwards. This is useful when a question gives us the result and asks which number was squared or cubed.

A cube uses three equal factors. Here are the cube facts worth having ready:

Zero and 1 are both squares and cubes: 0^² = 0^³ = 0, and 1^² = 1^³ = 1. Those belong to both lists too.

Following a chain of powers

To build the powers of 3, start at 3 and keep multiplying by 3: 3, 9, 27, 81, 243. Counting the factors gives 3^⁵ = 243.

1. 3^¹3
2. 3^²9
3. 3^³27
4. 3^⁴81
5. 3^⁵243

Each chain starts at index 1. The fourth entry in the powers of 3 is 81, so 81 = 3^⁴. Swapping the base and index gives a different calculation: 4^³ = 64.

Recognition isn't a new operation to memorise. If a fact won't come to mind, multiply by the base until you reach the number—or pass it.

Different powers, the same number

The number 64 appears in more than one chain. Six factors of 2 can be grouped into three pairs, each worth 4, or two groups of three, each worth 8.

All three names are true. If the question specifies a base, use that base: “64 as a power of 4” means 4^³.

Between neighbouring powers

Not every whole number is a square. Since 7^² = 49 and 8^² = 64, the number 50 lies between two neighbouring squares. There is no whole-number side length whose square is 50.

The same idea works for cubes. To decide whether 100 is a cube number, look for the whole-number cubes on either side of it.