Laws of Exponents for Real Numbers
Do you remember how to simplify the following?
(i)
(ii)
(iii)
(iv)
To get these answers, you would have used the following laws of exponents, which you have learnt in your earlier classes (Here a, n and m are natural numbers.
Remember a is called the base and m and n are the exponents.)
(i) | |
(ii) | |
(iii) | |
(iv) |
What is
it is
So you have learnt that
So, using (iii), we can get
We can now extend the laws to negative exponents too.
For example:
(i)
(ii)
(iii)
(iv)
Example 20: Simplify
How would we go about it?
It turns out that we can extend the laws of exponents that we have studied earlier, even when the base is a positive real number and the exponents are rational numbers.
We define for a real number a > 0 as follows:
Let a > 0 be a real number and n a positive integer. Then
In the language of exponents, we define
There are now two ways to look at
Therefore, we have the following definition:
Let a > 0 be a real number. Let m and n be integers such that m and n have no common factors other than 1, and n > 0. Then,
We now have the following extended laws of exponents: Let a > 0 be a real number and p and q be rational numbers.
Then, we have
(i) | (ii) |
(iii) | (iv) |