Repeated Factors and Powers

1. What are repeated factors?

You already know that in 4 × 7 = 28, the numbers 4 and 7 are factors of 28. Sometimes a product uses the same factor more than once. In 3 × 3 × 3 the factor 3 is used three times. We say 3 × 3 × 3 is a product of repeated factors, and each factor is 3.

You can see repeated factors with bottle tops (or counters, or bundle sticks). Put 3 bottle tops in a row. Now make 3 such rows: you have 3 × 3 = 9 tops in a square. Now make 3 of those squares: you have 3 × 3 × 3 = 27 tops. Each time, you multiplied by another 3. Step through the animation.

▶ Notice that each new picture is 3 copies of the one before. The number of 3s you multiply tells you how big the pile gets.

With bundle sticks the idea is the same: tie 2 sticks into a bundle, tie 2 bundles together, then tie 2 of those together. The last bundle holds 2 × 2 × 2 = 8 sticks, a product of three repeated factors of 2.

2. Writing repeated factors as a power

Instead of writing 2 × 2 × 2 × 2 × 2, we write the factor once and put a small number at its top right to show how many times it is used: $2\times 2\times 2\times 2\times 2=2^5$. This short form is called a power or index form. Its value is 32, so we say 32 is the fifth power of 2.

Every power has two parts. The big number at the bottom is the base: it is the repeated factor. The small raised number is the index (also called the exponent or power): it counts the factors. In $2^3$, the base is 2 and the index is 3, so $2^3=2\times 2\times 2$.

1. 1Check that every factor in the product is the same number. That number is the base.
2. 2Count how many times it appears. That count is the index.
3. 3Write the base, then write the index small and raised at its top right.

3. Squares, cubes and a big mistake to avoid

Two powers have special names. A number with index 2 is squared, because 3 × 3 bottle tops make a square. A number with index 3 is cubed, because 3 × 3 × 3 small blocks build a cube. So $6^2$ is "6 squared" and $4^3$ is "4 cubed". For other indices we say "to the power", e.g. $5^4$ is "5 to the power 4". Any number on its own has index 1: $8^1=8$.

4. Powers in everyday life

Powers appear whenever something is multiplied by the same number again and again. Fold a sheet of paper in half and you get 2 layers. Fold it again and you get 2 × 2 = 4 layers. Every fold doubles the layers, so after 5 folds you have $2^5$ = 32 layers.

▶ Watch the stack double with each fold. The number of folds becomes the index of 2.

• Chain messages: if each person forwards a message to 3 new people, round 4 reaches $3^4$ new people.
• Cocoa bags: a stack 4 bags long, 4 bags wide and 4 bags high holds 4 × 4 × 4 = $4^3$ bags.
• Square plots: a square farm plot 9 m by 9 m has an area of 9 × 9 = $9^2$ square metres.
• Place value: 1,000 = 10 × 10 × 10 = $10^3$, and one million = $10^6$.

Worked examples

Try each one yourself first, then open the solution and compare every step.