# Brush up your maths: Powers

## Raising to a power

We have already looked at raising to the power 2. For example, 52 = 5 × 5 = 25. We call this 5 to the power 2 or 5 squared. We call it 5 squared because it the area of a square the side of which is 5 units long.

We can raise to other powers. For example, 53 = 5 × 5 × 5 = 125. We call this 5 to the power 3 or 5 cubed. We call it 5 cubed because it the volume of a cube the side of which is 5 units long.

In the same way, 54 = 5 × 5 × 5 × 5 = 625. We call this 5 to the power 4 or 5 to the fourth. It has no geometrical name.

When we raise a number to a power, we multiply the power copies of the number together. When we have power = 1, we get the number itself: 51 = 5. We have one copy of the number multiplied.

### Exercise: raising to a power

What do we get if we raise 2 to the powers 1, 2, 3, 4, 5, and 6?

## Multiplying and dividing with powers

If we multiply two numbers both raised to powers together, we add the powers. For example, 32 = 3 × 3 = 9, 31 = 3. If we multiply 32 × 31 we get 3 × 3 × 3 = 27. We can also add the powers: 32 × 31 = 32+1 = 33 = 27. We are adding up the total number of 3s we will multiply.

In the same way, 42 × 43 = 42+3 = 45 = 1024. Compare 42 = 16, 43 = 64, 16 × 64 = 1024.

It has to be the same number we raise to each power. We cannot multiply 42 × 33 in this way.

If we divide two numbers both raised to powers, we subtract the powers. For example, 32 / 31 = 9/3 = 3. If we divide 32 / 31 we get (3 × 3) / 3 = 3. We can also subtract the second power from the first: 32 / 31 = 32–1 = 31 = 3.

### Exercise: multiplying and dividing with powers

What do we get if we multiply 63 by 62?

What do we get if we divide 63 by 62?

## Negative and zero powers

What would it mean to raise to the power zero? Any number raised to the power zero is one. For example, 50 = 1, 20 = 1, and 3450 = 1.

We can see this if we divide a number by itself. Clearly, 5 / 5 = 1. We can write 5 as 51. If we divide 5 by 5, we have 1 = 5 / 5 = 51 / 51 = 51–1 = 50. Hence 5 to the power zero = 1. In the same way, any number to the power zero = 1.

Now, we cannot explain raising to the power zero as multiplying no numbers together, it doesn't make sense. This is typical mathematicians' behaviour. The original definition of raising to a power applied only to powers which were positive whole numbers. We find out what raising to the power zero would have to be to be consistent with this original definition.

We do the same thing for negative powers.

1
5–1   =   –––
5

It is easy to see why this has to be the case. We divide 1 by 5. We can write 1 = 50 and 5 = 51. Then

1
––   =   50 ÷ 51   =   50–1   =   5–1
5

In the same way, 5–2 = 1/52 = 1/25 = 0.04, etc.

### Exercise: negative and zero powers

What do we get if we raise 2 to the powers 0, –1, –2, and –3?

What is 34 multiplied by 3–4?

What is 42 multiplied by 4–3?

## Fractional powers

What would it mean to raise to the power 1/2? Raising to the power 1/2 means we find the square root, the number which multiplied by itself gives our original number. This another example of mathematicians extending things.

We can see why this is if we multiply 3½ by 3½. We get 3½ × 3½ = 3½+½ = 31 = 3. So 3½ must be the square root of 3.

In the same way, raising to the power 1/3 gives the cube root, where multiplying three of these together gives the original number, raising to the power 1/4 gives the fourth root, where multiplying four of these together gives the original number, and so on.

We can raise to any power. Raising to the power 2/3 would give the cube root of the number squared, for example. Raising to the power –½ would the reciprocal of the square root.

### Exercise: fractional powers

What is 9 raised to the power ½?

What is 8 raised to the power –1/3?

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This page maintained by Martin Bland.
Last updated: 26 November, 2007.