How to Multiply and Divide Exponents?

Exponents and powers are used to simplify the representation of very large or very small numbers. Power is a number or expression that represents the repeated multiplication of the same number or factor. The value of the exponent is the number of times the base is multiplied by itself.

Multiplying and dividing exponents involves specific rules that simplify calculations involving powers. We will discuss these rules in the article below.

Example for exponents:

If we need to express 3 × 3 × 3 × 3 × 3 in a simple way, we may write it as 3⁵, where 3 is the base and 5 is the exponent. The entire expression 3⁵ is considered to represent power. Example for powers: 5³ = 5 raised to power 3 = 5 × 5 × 5 = 125, 6⁴ = 6 raised to power 4 = 6 × 6 × 6 × 6 = 1296. A number's exponent represents the number of times the number has been multiplied by itself. 3 is multiplied by itself for n times, 3 × 3 × 3 × 3 × …n times = 3ⁿ. 3ⁿ is an abbreviation for 3 raised to the power of n. As a result, exponents are sometimes known as power or, in certain cases, indices.

General Form of Exponents 

The exponent indicates how many times a number should be multiplied by itself to obtain the desired results. As a result, any number 'b' raised to the power 'p' may be expressed as :

b^p =  {b × b × b × b × ...  × b} p times

Here b is any number, and p is a natural number.

• Here, b^p is also called the p^th power of b.
• ‘b’ represents the base, and ‘p’ is the exponent or power.
• Here ‘b’ is multiplied ‘p’ times, and thereby exponentiation is the simplified method of repeated multiplication.

Some Basic Rules of Exponents 

1. Product Rule ⇢ aⁿ × a^m = a^{n + m}
2. Quotient Rule ⇢ aⁿ / a^m = a^{n - m}
3. Power Rule ⇢ (aⁿ)^m = a^{n × m} or ^m√aⁿ = a^n/m
4. Negative Exponent Rule ⇢ a^-m = 1/a^m
5. Zero Rule ⇢ a⁰ = 1
6. One Rule ⇢ a¹ = a

How to Multiply and Divide Exponents?

Multiplying and Dividing Exponents involves addition and subtraction of powers respectively if the base is same.

To Multiply Exponents 

First Case: When Multiply exponents with the same Base 

According to this rule: The product of two exponents with the same base but distinct powers equals the base raised to the sum of the two powers or integers; this is also known as the Multiplication Law of Exponents. When multiplying two expressions with the same base, we can use, 

mⁿ₁ × mⁿ₂ = m⁽ⁿ₁ ^{+ n}₂⁾  

Where m is the common base and n₁ and n₂ are the exponents.

For Example, Multiply 3³ × 3⁶?

Given: 3³ × 3⁶ 

Here bases are same. So we will use: mⁿ₁ × mⁿ₂ = m⁽ⁿ₁ ^{+ n}₂⁾  

Therefore, = 3 ⁽³⁺⁶⁾

= 3⁹ 

Second Case: When Multiply Exponents with a different bases

When there is different base with same exponents , we will use the formula : 

m^p × n^p = (m × n)^p. 

Here m and n are the different bases and  p is the exponent.

Example: Multiply 2³ × 4³

Given: 2³ × 4³ 

Here, we will use: m^p × n^p = (m × n)^p

= (2 × 4)³

= 8³               

In these ways in different cases we can divide and multiply Exponents.

To Divide Exponents

The laws of exponents simplify the process of simplifying expressions. When dividing exponents with the same base, the basic rule is to subtract the given powers. This is also known as the Division Law or  Exponent Quotient Property.

mⁿ₁ ÷ mⁿ₂ = mⁿ₁/ mⁿ₂ = m⁽ⁿ₁ ^{- n}₂⁾

First Case: Dividing Exponents with the Same Base 

We utilize the basic rule of subtracting the powers to divide exponents with the same base. Consider the expression mⁿ₁ ÷ mⁿ₂, where  'm' is the common base and the exponents 'n₁' and 'n₂' are the exponents. According to the 'Quotient property of Exponents,'

mⁿ₁ ÷ mⁿ₂ = mⁿ₁/ mⁿ₂ = m⁽ⁿ₁ ^{- n}₂⁾  

Example: Divide 3⁵ ÷ 3³ 

Here as we can see bases are same but different powers .

So the division law or Quotient law  : mⁿ₁ ÷ mⁿ₂   =  mⁿ₁/ mⁿ₂ = m ⁽ⁿ₁ ^{- n}₂⁾  

Here, 3⁵ ÷ 3³

= 3⁵/3³

= 3^(5-3)

= 3²   

Second Case: Dividing Exponents with different Bases 

We apply the 'Power of quotient property' to divide exponents with different bases and the same exponent, which is

(m/n)^p = m^p/n^p  

Consider the formulas m^p ÷ n^p, which has distinct bases but the same exponent.

Example: Divide: 15³ ÷ 3³.

This can be solved using the 'Power of quotient property' as,

(m/n)^p = m^p/n^p.

= 15³ ÷ 3³

= (15 / 3)³

= 5³.    

Sample Questions

Question 1: Simplify or Divide 25⁴/5⁴  

Solution: 

Here bases are different with same Exponent,

We will use the formula, (m/n)^p = m^p/n^p  

Therefore, = 25⁴/5⁴

= (25/5)⁴

= 5⁴

= 625

Question 2: Find the value of the expression, 15⁸ × 15³

Solution:

Given: 15⁸ × 15³

When multiplying two expressions with the same base but different exponent, 

mⁿ₁ x mⁿ₂ = m⁽ⁿ₁ ^{+ n}₂⁾ formula, where m is the common base and n₁ and n₂ are the exponents.

By Applying this rule, 

we get, = 15⁸  × 15³

= 15^{(8 + 3)}

= 15¹¹

Question 3: What is the product of (2x³y⁵ ) and (3x⁴y²)?

Solution:

The product of  (2x³y⁵) and (3x⁴y²)

= (2x³y⁵) × (3x⁴y²)

= (2 × 3) × x³x⁴ × y⁵y²                    

When multiplying two expressions with the same base, we can use mⁿ₁ × mⁿ₂ = m⁽ⁿ₁ ^{+ n}₂⁾ formula, where m is the common base and n₁ and n₂ are the exponents.

= 6x³⁺⁴ × y⁵⁺²

= 6x⁷y⁷ 

Question 4: What is x³ divided by x²?

Solution:

Here given: x³divided by x² 

here bases are same but exponents are different,

So we use the division law or Quotient law: mⁿ₁ ÷ mⁿ₂   =  mⁿ₁/ mⁿ₂ = m ⁽ⁿ₁ ^{- n}₂⁾  

So write it as x³/x²

= x^{3 - 2}

= x¹

= x

Question 5: Evaluate a³ × a⁵ × a^-6 

Solution:

Given that: a³ × a⁵ × a^-6 

Here bases are same but exponents are different ,By using product rule or multiplication law .

mⁿ₁ × mⁿ₂ = m⁽ⁿ₁ ^{+ n}₂⁾

= a³ × a⁵ × a^-6

= a^{(3 +5)} × a^-6

= a⁸ × a^-6

= a{8+ (-6)} {Using by product rule}

= a^8-6

= a² 

Question 6: Divide 10⁵/5⁵

Solution:

Here bases are different with same Exponent ,

we will use the formula  : (m/n)^p = m^p/n^p  

Therefore, = 10⁵/5⁵

= (10/5)⁵

= 5⁵

= 3125