# A Key To The Laws Of Exponents

Can you read 10,000,000,000,000,000,000,000? This massive natural number is 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.

**Learn:** Exponent

## Rules of Exponents With Examples

Exponents are defined as a number that tells how many times we have to multiply the base number. It is written above the right side of the base number.

1. 5^{2} = “5 raised to the power of 2” or “5 squared.”

2. 5^{3} = “5 raised to the power of 3” or “10 cubed.”

Example 1 :10,000 = 10 x 10 x 10 x 10 = 10^{4}

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} × a^{q} = a^{(p+q)}

a = base : p,q = exponents

Example 1: Let us calculate,

3^{2}×3^{4}

Solution:

3^{2} × 3^{4} =3^{{(2+4)}} = 3^{6}

In the above example, the base numbers are the same. (i.e.,) 3^{2} and 3^{4}. 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)}} i.e. 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 bases are the same. (i.e., 3^{4} and 3^{2}).

**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

**Example 1: **

Multiplying powers with the same base

3^{2} × 3^{3} =?

**Solution:**

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

So,

3^{2} × 3^{3} = 3^{{(2+3)}} = 3^{5}

**Example 2: **

Dividing powers with the same base

4^{5} ÷ 4^{2}

**Solution:**

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}

**Example 3:**

What is the value of 2 power 9 and 2 power 7?

**Solution:**

2 power 9 can be represented as 2^{9}.

The value of 2 power 9 = 2^{9} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512

The value of 2 power 7 = 2^{7} = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

### Practice Problems

- Find the value of (4
^{3})^{2}. - What is the value of 6
^{-2}. 6^{5}? - Evaluate: (0.5)
^{2} - Find the value of the product of 2 power 7 and 2 power 9.

Thus, we have seen this basic introduction and examples of the laws of exponents. For a complete understanding of the topic, please visit our site or download the BYJU’S – The Learning App.