# Laws of Exponents for Real Numbers

**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 **

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)172 · 175** =

**(ii) ** =

**(iii)** =

**(iv)7−3 · 9 −3** =

**Example 20: Simplify**

**Instruction**

**(i)**2 2 3 × 2 1 3

**(ii)**3 1 5 4

**(iii)**7 1 5 7 1 3

**(iv)**13 1 5 × 17 1 5

**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 n a =b, if bn = a and b > 0.**

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) |