# Laws of Exponents for Real Numbers

Complete above steps to enable this content

Do you remember how to simplify the following?

**(i) 172 · 175**=

**(ii) ** =

**(iii)** =

**(iv) 73 · 93** =

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

Complete above steps to enable this content

**Suppose we want to do the following computations:**

**(i)**2 2 3 · 2 1 3

3 1 5 4

- we have exponent is (
a m a n = a m − n , m > n ) - calculate the exponent is
then we get it is7 1 x 3 − 1 x 5 5 x 3 - Hence it is
7 − 2 15

Complete above steps to enable this content

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. (Later you will study that it can further to be extended when the exponents are real numbers.) But before we state these laws, and to even make sense of these laws, we need to first understand what, for example

We define

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) |