# Exponents

Exponents are the power of any number or variable.

Example:- 2⁻², 4⁶, a⁴, x⁷ & 5⁻³

Where,

2, 4, a, x & 5 are the base.

While -2, 6, 4, 7 & -3 are the exponents.

## What are the Laws of Exponents?

Laws of exponents are used to solve the problems where integers have powers.

Different types of Laws of Exponents are:-

### List of the Laws of Exponents-

**a^m × a^n = a^m+n**

**a^m ÷ a^n = a^m-n**

**a^m × b^m = (ab)^m**

**a^m ÷ b^m = (a÷b)^m**

**(a^m)ⁿ = (a)^mn**

### 1. Laws of exponents for multiplication

a^m × a^n = a^m+n

Here we have a formula of exponents for multiplication-

Where,

a is the base, m & n are the exponents.

It is used where the bases are the same.

How to use the law of exponents for multiplication, understand with the example:-

**Evaluate 2⁻⁷ × 2⁸**

Solve:-

= 2⁻⁷ × 2⁸

{We will use the law of exponents for multiplication for the same base}

= a^m × a^n = a^m+n

= 2⁻⁷ × 2⁸ = 2⁻⁷⁺⁸

= 2¹

Here we get our answer 2¹.

### 2. Laws of exponents for the division

a^m ÷ a^n = a^m-n

Here we have a formula of exponents for division-

Where,

a is the base, m & n are the exponents.

It is used where the base is the same.

How to use the law of exponents for division, understand with the example:-

**Evaluate 4⁵ ÷ 4⁻¹**

Solve:-

= 4⁵ ÷ 4⁻¹

{We will use the law of exponents for the division of the same base}

= a^m ÷ a^n = a^m-n

= 4⁵ ÷ 4⁻¹ = 4⁵⁻⁽⁻¹⁾

= 4⁵⁺¹

= 4⁶

Here we get our answer 4⁶.

### 3. Laws of exponents for different bases & same powers

a^m × b^m = (ab)^m

Here we have a formula of exponents for different bases & same powers-

Where,

a & b are the bases, m is the exponent.

It is used where the bases are different & it works in multiplication.

How to use the law of exponents for different bases, understand with the example:-

**Evaluate 4⁵ × 5⁵**

Solve:-

= 4⁵ × 4⁵

{We will use the law of exponents for multiplication of the same base}

= a^m × b^m = (ab)^m

= 4⁵ × 5⁵ = (4×5)⁵

= (4×5)⁵

= 20⁵

Here we get our answer 20⁵.