Properties of Real Numbers - Negative Exponents

What do we mean when we say "3 raised to -2"? Now to understand the meaning of negative exponents, we need to stop thinking that negative exponent means "repeated multiplication" because it is not. Negative exponent has a different "definition", and that is what we are going to discuss in this video.

Now before we define negative exponent, we have the following prerequisites:
First, we need to know the meaning of the "positive integer" exponent.
Second, we need to know the "product of powers" property.
Third, we need to know the meaning of the "zero" exponent.
Once we understand all of these three things, we will understand the logic of why negative exponents are defined the way it is.

Now let's review the meaning of positive integer exponent. When the exponent is a positive integer, exponentiation is defined as "repeated multiplication". The expression "b raised to n" means "n copies of b multiplied together".

So for example,
2 raised to 3 equals 2 x 2 x 2 equals 8
3 raised to 5 equals 3 x 3 x 3 x 3 x 3 = 243
7 raised to 1 equals 7
(1/2) raised to 2 equals 1/2 x 1/2 equals 1/4
(-2) raised to 4 equals (-2) x (-2) x (-2) x (-2) = 16

Now using this definition, we will prove one property of exponents which is the Product of Powers.
Let's say we have 'b raised to m' times 'b raised to n'; that is, two exponential expressions with the same base multiplied together.
If the exponents are positive integers, then by definition this means m copies of b multiplied together times n copies of b multiplied together.
So in total, how many copies of b are there? There are m plus n copies of b multiplied together. So if there are m plus n copies of b multiplied together, then by definition, that means 'b is raised to m+n'. This leads us to the following property of exponents:

The product of two exponential expressions with the same base
is equal to the base raised to the sum of the exponents.
'b raised to m' times 'b raised to n' equals 'b raised to m+n'.

Now to show this, let's evaluate '2 raised to 4' times '2 raised to 2'.
This is equal to 'four twos multiplied together' times 'two twos multiplied together'.
Now this is equal to 16 times 4.
And 16 times 4 is 64.

Now let's evaluate '2 raised to 4+2' or '2 raised to 6'.
This equals six twos multiplied together, which is equal to 64.

Now using the Property of Product of Powers, we can define the meaning of zero exponent.

Let's say we have '3 raised to 0' times '3 squared'.
Now by property of product of powers, this is 3 raised to 0+2.
0 plus 2 is 2, so we have '3 squared'.
3 squared is equal to 3 times 3, and 3 times 3 is 9.

So, therefore, '3 raised to 0' times '3 squared' is equal to 9, or '3 raised to 0' times 9 equals 9.
But for this equation to be true, then what must be the value of '3 raised to 0' so that when multiplied by 9, gives 9?
That value is 1 because any number multiplied by 1 equals the number.
So, therefore, '3 raised to 0' must be equal to 1.

This leads us to the following definition:

Any non-zero number "b" raised to 0 is equal to 1.
For example,

2 raised to 0 equals 1.
Negative one-third raised to 0 equals 1.
0.67 raised to 0 equals 1.
5/7 raised to 0 equals 1.
Negative 6 raised to 0 equals 1.
But take note that 0 raised to 0 is undefined.

Now using the definition of zero exponent, we can define the meaning of negative exponent.

Let's say we have '5 cubed' times '5 raised to negative 3'.
By Property of Product of Powers, this equals 5 raised to 3-3.
3 minus 3 is 0, so we have '5 raised to 0'.
And we know that '5 raised to 0' is 1.

So therefore, '5 cubed' times '5 raised to negative 3' is equal to 1.
But for this equation to be true, then what must be the value of '5 raised to negative 3' so that when multiplied with '5 cubed', gives 1?
That value is the reciprocal of '5 cubed' which is '1 / (5 cubed)'
because any non-zero number multiplied by its reciprocal equals 1.
So therefore, '5 raised to -3' must be equal to '1 / (5 cubed)'.

This leads us to the following definition:

Any non-zero number 'b' raised to '-n', where n is a positive integer, is equal to '1 over b raised to n'.

For example,
3 raised to negative 1 equals '1 over (3 raised to 1)' equals one-third.
2 raised to negative 3 equals '1 over (2 raised to 3) equals '1 over 8'.
Negative 5 raised to negative 2 equals '1 over (the square of negative 5)' equals '1 over 25'.
But take note that 0 raised to a negative integer is undefined.

Now I hope this helped you understand the meaning of negative exponents. Again, do not think of it as "repeated multiplication" because that's impossible. Think of it as a "definition", because it is actually a definition.

Thanks for watching.