# Laws of Exponents: Rules of Exponents with Examples: Product with the Same Bases

Doorsteptutor material for NDA Mathematics is prepared by world's top subject experts: fully solved questions with step-by-step explanation- practice your way to success.

There are in general six **laws of exponents** in Mathematics. Exponents are used to show, repeated multiplication of a number by itself. For example, can be represented as . Exponent is ‘3’ which stands for the number of times the number is multiplied. 7 is the base here which is the actual number that is getting multiplied. So basically, exponents or powers denotes the number of times a number can be multiplied. If the power is 2, that means the base number is multiplied two times with itself. Some of the examples are:

A number ‘a’ is multiplied by itself n-times, then it is represented as where a is the base and n is the exponent.

Exponents follow certain rules that help in simplifying expressions which are also called its laws. Let us discuss the laws of exponents in detail.

## Rules of Exponents with Examples

Six laws or rules defined for exponents. In the table below, all the laws are represented.

Now let us discuss all the laws one by one with examples here.

### Product with the Same Bases

As per this law, for any non-zero term a,

where m and n are real numbers.

**Example 1: What is the simplification of** **?**

Solution:

Here given,

Here 4 no power is 5 means 4 is multiplied four times, and other 4^{th} power is 1.

Here product with the same base rule applied, sum of the powers.

Sum of 4 and 1.

**Example 2: What is the simplification of** **?**

Solution:

Here given,

Here given power is in term of negative, but same rule applied of product with same base.

Sum of the and .

Sum of two negative numbers is also negative.

**Note:** We can state that the law is applicable for negative terms also. Therefore, the term m and n can be any integer.

### Quotient with Same Bases

As per this rule,

where a is a non-zero term and m and n are integers.

**Example 3: Find the value when** **is divided by**

Solution: As per the question,

Here quotient with same bases rule applied,

Here base 12 and its powers are and .

Here subtract of and

power is we can write in this form,

### Power Raised to a Power

According to this law, if ‘a’ is the base, then the power raised to the power of base ‘a’ gives the product of the powers raised to the base ‘a’, such as,

where a is a non-zero term and m and n are integers.

**Example 4: Express** **as a power with base** **.**

Solution: We have,

We have to write as a power with base 2.

So,

Here power raised to a power rule is applied.

Therefore, multiplication of powers.

### Product to a Power

As per this rule, for two or more different bases, if the power is same, then,

where a is a non-zero term and n is the integer.

**Example 5: Simplify and write the exponential form of:**

Solution: Here given,

We can write,

Therefore,

Here power is same but base is not.

So, product to a power rule is applied.

### Quotient to a Power

As per this law, the fraction of two different bases with the same power is represented as,

where a and b are non-zero terms and n are an integer.

**Example 6:** Simplify the expression and find the value:

Solution:

Here quotient to a power rule is applied.

We can write the given expression as,

Divide 15 by 5.

3^{rd} power is 3.

So, 3 is multiplied by three times.

### Zero Power

According to this rule, when the power of any integer is zero, then its value is equal to 1, such as,

where ‘a’ is any non-zero term.

**Example 7: What is the value of**

Solution:

Here given expression,

Here zero power rule is applied.