Åk 6–9

6.1 From Multiplication to Powers

If the base is positive
Image that you are going to multiply a number by itself many times.

Example: 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = ?

For practical reason, mathematicians have decided to write this an easier way.  You then use something called powers. Powers can then be described of as product of identical factors.

The example above would then be written as follow:
3 · 3 · 3 · 3 · 3 = 243.

The power 35 is read as ”three to the power of five”. This is because we have multiplied the number 3 five times with itself.

23 = 8 because 2 · 2 · 2 = 8. Here the two is called the base and the three for the exponent We take for example: 7 · 7 · 7 · 7 = 74  If the base is negative

The powers can also be expressed with negative bases and exponents.  Here are a few examples with negative bases. Can you solve them?

(-3)2 = ?
(-5)4 = ?
(-2)3 = ?

This is how you do it:
(-3)2 = (-3) ∙ (-3) = 9
(-5)4 = (-5) ∙ (-5) ∙ (-5) ∙ (-5) = 25 ∙ 25 = 625
(-2)3 = (-2) ∙ (-2) ∙ (-2) = -8

On many calculators there is an exponent function. Foto: Fredrik Enander

If the exponent is positive

An exponent, which is a positive number, indicates how many times the base should be multiplies by itself.

For example, 24 means that the base 2 should be multiplied by itself 4 times (2 · 2 · 2 · 2).

If the exponent is negative
Maybe you know that 10-2
is the same thing as 0.01?  It is not quite as simple to explain the connection between the exponent -2 and the number 0.01 in this example.  It can be describes like this:

0.01 is the same thing as a hundredth.  If we write it as a fraction, it looks like this:

 0.01 = 1 100 = 1 10 · 10 = 1 102

We can then make the following conclusion:

 10-2 = 1 102

We can also show this by using the rules of expontents (see more on page 6.5 Rules of Exponents). The rules for exponent division says that:

 ab ac = ab-c

Example

 Carry out the division 74 76 .

If we use the rules of exponents we get:

 74 76 = 7-2

According to the definition of powers:

 74 76 = 7 · 7 · 7 · 7 7 · 7 · 7 · 7 · 7 · 7 = 1 7 · 7 = 1 72

 This motivates that 7-2 = 1 72 .

The example above allows us to create following definition of powers with negative exponents:

For all numbers a (except a = 0) and all whole numbers n:

 a-n = 1 an