We have already looked at raising to the power 2.
For example, 5^{2} = 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, 5^{3} = 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, 5^{4} = 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:
5^{1} = 5.
We have one copy of the number multiplied.

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

Check answer to raising to a power exercise.

In the same way, 4^{2} × 4^{3} = 4^{2+3} = 4^{5} = 1024.
Compare 4^{2} = 16, 4^{3} = 64, 16 × 64 = 1024.

It has to be the same number we raise to each power.
We cannot multiply 4^{2} × 3^{3} in this way.

If we divide two numbers both raised to powers, we subtract the powers.
For example, 3^{2} / 3^{1} = 9/3 = 3.
If we divide 3^{2} / 3^{1} we get (3 × 3) / 3 = 3.
We can also subtract the second power from the first:
3^{2} / 3^{1} = 3^{2–1} = 3^{1} = 3.

What do we get if we multiply 6^{3} by 6^{2}?

What do we get if we divide 6^{3} by 6^{2}?

Check answer to multiplying and dividing with powers exercise.

We can see this if we divide a number by itself.
Clearly, 5 / 5 = 1.
We can write 5 as 5^{1}.
If we divide 5 by 5, we have 1 = 5 / 5 = 5^{1} / 5^{1} =
5^{1–1} = 5^{0}.
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 = 5^{0} and 5 = 5^{1}.
Then

1

–– = 5^{0} ÷ 5^{1} =
5^{0–1} = 5^{–1}

5

In the same way, 5^{–2} = 1/5^{2} = 1/25 = 0.04, etc.

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

What is 3^{4} multiplied by 3^{–4}?

What is 4^{2} multiplied by 4^{–3}?

Check answer to negative and zero powers exercise.

We can see why this is if we multiply 3^{½} by 3^{½}.
We get 3^{½} × 3^{½} =
3^{½+½} = 3^{1} = 3.
So 3^{½} 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.

What is 9 raised to the power ½?

What is 8 raised to the power –1/3?

Check answer to fractional powers exercise.

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Last updated: 26 November, 2007.