Can you read 10,000,000,000,000,000,000,000? These massive natural numbers are not easy to read, recognize and evaluate. Exponents make it easy to read, recognize and evaluate very large numbers. Exponents are also called powers or indices. What is meant by exponents? What are the laws of exponents? How to apply the laws of exponents to simplify expressions? Let us take an overview of the laws of exponents.

Exponents are defined as a small number written with a large number to illustrate that it has been raised to a power. The process of exponents is called as “raised to the power”. In exponents, the following two specialized names are available.

1. 52 = “5 raised to the power of 2” or “10 squared”

2. 53 = “5 raised to the power of 3” or “10 cubed”

Example1 :10,000 = 10 x 10 x 10 x 10 = 104

The laws of exponents state the following rules to simplify the expressions. Some of them are as follows:

**Rule 1** : When the numbers having the same base are multiplied, add the exponents.

\(a^p~\times~a^q\) = \(a^{(p+q)}\)

\(a\) = base :\(p,q\) = exponents

Example 1: Let us calculate,

\(3^2~\times~3^4\)

Solution:

\(3^2~\times~3^4\) =\(3^{(2+4)}\) =\(3^6\)

In the above example, the base in

\(3^2\) and\(3^4\) is same. The sum of the powers is 6.

**Rule 2**: When the numbers having the same base are divided, subtract the exponents..

\(a^p ÷ a^q\) =\(a^{(p-q)}\)\(a\) = base :\(p,q\) = exponents

Example 2:\(3^4 ÷ 3^2\) = ?

Solution:

\(3^4 ÷ 3^2\) =\(3^{(4-2)}\)=\(3^2\)

In the above example, the base in

\(3^4\) and\(3^2\) is same.

**Rule 3**: Multiply the powers when the numbers are raised by another number.

\((a^p)^q\) =\(a^{(pxq)}\) =\(a^{(pq)}\)

Example 3 :

\((2^3)^2\) =?

Solution:

\((2^3)^2\) =\(2^{(3×2)}\) =\(2^6\)

**Example problems**

** Multiplying powers with the same base**

**1)**\(3^2~\times~3^3\) = ?

Solution:

\(3^2\) =\(3~\times~3\)

\(3^3\) =\(3~\times~3~\times~3\)

\(3^2~\times~3^3\) =\((3~\times~3)~\times~(3~\times~3~\times~3)\) =\(3~\times~3~\times~\times~3~\times~3~\times~3\)

Answer=

\(3^5\)

In the above example, as per rule 1, add the powers when the numbers are multiplied.

So,

\(3^2~\times~3^3\) =\(3^{(2+3)}\) =\(3^5\)

Dividing powers with the same base

**2)**

\(4^5 ÷ 4^2

\) =?

Solution:

\(4^5\) =\(4~\times~4~\times~4~\times~4~\times~4\)

\(4^2\) =\(4~\times~4\)\(4~\times~4~\times~4~\times~4~\times~4\)

\(4^5 ÷ 4^2\) =\(4~\times~4\)=\(4~\times~4~\times~4\)

Answer=

\(4^3\)

In the above example, as per rule 2, subtract the powers when the numbers are divided.

So,

\(4^5 ÷ 4^2\) =\(4^{(5-2)}\) =\(4^3\)

Taking power of a power

**3)**\((2^2)^3\)

Solution:\((2^2)^3\) =\(2^2~\times~2^2~\times~2^2\)

As per rule 1, add powers,

=\(2(2+2+2)\)

Answer = 26

In the above example, as per rule 3, multiply the powers when the numbers are raised by another number. So,

\((2^2)^3\) =\(2^{(2×3)}\) =\(2^6\)

We have thus seen this basic introduction and examples of the laws of exponents.For the complete understanding of the topic please visit our site or download the byjus learning app.