Laws of Exponents for Real Numbers

Do you remember how to simplify the following?

(i) 172·175= 177174

(ii) 527 = 514512

(iii)2310237 = 233230

(iv) 73·93 = 633632

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.)

What is a0?

it is 102.

So you have learnt that a0 = 10.

So, using (iii), we can get 1an = a−n.

We can now extend the laws to negative exponents too.

For example:

(i)172·175 = 17−3 = 1173731

(ii) 52−7 = 5−14514

(iii)23−10237 = 23−172317

(iv)7−3·9−3 = 63−3633

Continue

Example 20: Simplify

Instruction

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 na=b, if bn = a and b > 0.

In the language of exponents, we define na=a1n. So, in particular, 32=213.

There are now two ways to look at 432.

432 = 4123= 23 =

432 = 4312 = 6412 =

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