18.2 Powers of numbers

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.

blackboard-1644744_640

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 × am = 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/am = a(nm).

Notice that

1 = an/an = a(nn) = 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.

a1/2 × a1/2 = a1 = a.

So a1/2 is the number which when multiplied by itself gives a. We say that a1/2 is the square-root of a. For example, 91/2 = 3 because 3 × 3 = 9. Similarly

a1/3 × a1/3 × a1/3 = a1 = a.

We say that a1/3 is the cube-root of a. For example, 271/3 = 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.

table 1 cropped

Can we write numbers like 2, 3, 4, 5…., whose values lie between 1 and 10, in the form 10x, 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.

table 2 cropped

You can see that this works because

2 × 5 = 100.3010 × 100.6990 = 10(0.3010 + 0.6990) =101 = 10.

And

8/2 = 100.9031/100.3010 = 10(0.9031 – 0.3020) = 100.6021 = 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

 

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