We have met examples of this topic in several previous posts, especially post 16.7. Now I want to write about it in more detail.

If we want to write a number a multiplied by itself 5 times, we could write a × a × a × a × a but this is very tedious – especially when we multiply the number by itself very many times. So we define a⁵ as being a multiplied by itself 5 times:

a⁵ = a × a × a × a × a.

We say that a is raised to the power 5. We usually call a raised to the power 2 “a squared” and a raised to the power 3 “a cubed”.

What happens when we multiply a² by a³? The result is a multiplied by itself 5 times which is a⁵:

a² × a³ = (a × a) × (a × a × a) = a × a × a × a × a = a⁵.

From this example, we can see that, in general

aⁿ × a^m = a^{(n + m)}

where n and m represent any numbers.

Now let’s think about a⁵ divided by a²:

a⁵/a² = (a × a × a × a × a)/( a × a) = a × a × a = a³.

From this example, we can see that, in general

aⁿ/a^m = a^{(n – m)}.

Notice that

1 = aⁿ/aⁿ = a^{(n – n)} = a⁰.

So any number raised to the power 0 is equal to 1.

Powers need not be integers (whole numbers) – they can also be fractions.

a^1/2 × a^1/2 = a¹ = a.

So a^1/2 is the number which when multiplied by itself gives a. We say that a^1/2 is the square-root of a. For example, 9^1/2 = 3 because 3 × 3 = 9. Similarly

a^1/3 × a^1/3 × a^1/3 = a¹ = a.

We say that a^1/3 is the cube-root of a. For example, 27^1/3 = 3 because 3 × 3 × 3 = 27.

All this becomes a lot simpler when we raise the number 10 to different powers, because 10 is the base of our conventional counting system. This is illustrated in the table below.

Can we write numbers like 2, 3, 4, 5…., whose values lie between 1 and 10, in the form 10^x, where x is a number? Yes, we can, but it is not easy to calculate the value of x. Some examples are given in the table below.

You can see that this works because

2 × 5 = 10^0.3010 × 10^0.6990 = 10^{(0.3010 + 0.6990)} =10¹ = 10.

And

8/2 = 10^0.9031/10^0.3010 = 10^{(0.9031 – 0.3020)} = 10^0.6021 = 4.

We will use these ideas in my next post – on logarithms.

 

Related posts 

17.19 Calculating distances from speeds – integration
17.11 Measuring angles
17.4 Displacement, velocity and acceleration
17.2 Vectors
16.50 Directions of forces
16.7 Writing numbers
16.6 Exponential decay
16.5 Exponential growth