Multiplying and Dividing Exponents

An exponent tells us how many times a number (called the base) is multiplied by itself. Multiplying and dividing exponents involves addition and subtraction of powers, respectively, if the base is the same.

Apart from addition and subtraction of exponents, we can also use simple laws to simplify expressions quickly.

Solved Examples

Question 1: 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 

Question 2: Multiply 2³ × 4³

Given: 2³ × 4³ 

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

= (2 × 4)³

= 8³          

Question 3: 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   

Question 4: 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.    

Question 5: Simplify or Divide 25⁴/5⁴  

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 6: Find the value of the expression, 15⁸ × 15³

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

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 8: What is x³ divided by x²?

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 9: Evaluate a³ × a⁵ × a^-6 

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 10: Divide 10⁵/5⁵

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

= 2⁵

= 32