Multiplying and Dividing Exponents Lesson: Rules & Examples

Lesson Overview

• What Are Exponents?
• How to Divide Exponents
• How to Divide Exponents With Variables
• How to Multiply Exponents
• Examples of Multiply and Divide Exponents

Multiplying and dividing exponents makes math easier by helping us work with very large or very small numbers. It shows simple rules for dealing with exponents during multiplication and division.

What Are Exponents?

Exponents are a quick and easy way to show repeated multiplication of the same number or variable. It's like a shortcut for writing long multiplication. 

Above in the expression 3⁴, the big number (base) is 3, and the small number (exponent) is 4. 

For example: Instead of writing 3 x 3 x 3 x 3, we can write 3⁴. The exponent ⁴ indicates that the base 3 should be multiplied by itself four times.

3⁴ (read as "3 to the power of 4") means 3 x 3 x 3 x 3 = 81

How to Divide Exponents

We follow specific rules when dividing exponents, depending on whether the bases are the same. Division simplifies the repeated multiplication involved into smaller, more manageable steps.

• Example: 5⁵ ÷ 5² = 5³  
• (5 x 5 x 5 x 5 x 5) ÷ (5 x 5) = 5 x 5 x 5 = 125

Rules for Dividing Exponents

Take this quiz :

Multiplying and Dividing Exponents quiz

How to Divide Exponents With Variables

Dividing exponents with variables might sound tricky, but it's easier than you think! 

Dividing Exponents with Same Base

When dividing exponents with the same base, simplify by subtracting the denominator's exponent from the numerator's. The base remains the same.

Example:

x⁵ / x³ = x⁽⁵⁻³⁾ = x²

This rule applies to any base, not just x. Simplify the expression by subtracting the exponents.

Dividing Exponents with Different Bases

Simplifying exponents with different bases is only possible if the exponents are identical. Therefore, if the bases are different, the exponents must be the same to simplify the expression.

Example: 

2⁴ / 3² cannot be simplified further.

To simplify an expression, you can only subtract exponents when the bases are the same. Since the bases here (2 and 3) are different, we can't simplify this expression any more.

Dividing Exponents With Variables

When dividing exponents with variables, follow the same rules as with numerical bases:

• Same Base: Same Base: If the bases are the same, just subtract the exponents! 
• Different Bases: You can't simplify unless the exponents are the same. 

Example:

a⁵b³ / a²b = a⁽⁵⁻²⁾b⁽³⁻¹⁾ = a³b²

Imagine you have a bunch of balloons. Some are red (a) and some are blue (b). You can write this as a⁵b³ which means you have 5 red balloons and 3 blue balloons.

Now, some of your balloons fly away! You lose 2 red balloons (a²) and 1 blue balloon (b). You can write this as a²b.

To find out how many balloons you have left, you can use a cool trick with exponents:

a⁵b³ / a²b = a⁽⁵⁻²⁾b⁽³⁻¹⁾ = a³b²

This is like saying:

• Start with 5 red balloons, lose 2, and you have 3 left (5 - 2 = 3)
• Start with 3 blue balloons, lose 1, and you have 2 left (3 - 1 = 2)

So you end up with 3 red balloons (a³) and 2 blue balloons (b²). 

Take this quiz :

Arithmetic Challenge: Multiplying and Dividing Fractions Quiz

How to Multiply Exponents

When you multiply exponents, you're adding the powers of the same base number! It's a quick way to simplify big math problems.

Rules for Multiplying Exponents

Multiplying exponents is all about using three simple rules to make expressions with exponents easier to solve. Check out the table below to see how it works!

Multiplying Exponents with the Same Base

When you multiply exponents with the same base, you simply add the exponents together.

• Example:  x² * x³ = x^(2+3) = x⁵
  • That means, when you have x raised to the power of 2, and you multiply it by x raised to the power of 3, you just add the exponents together (2 + 3) to get x⁵!

Multiplying Exponents with Different Bases

If the bases are different, you can't just add or subtract the exponents. Instead, you need to solve each part of the problem separately and then multiply the answers you get.

• Example: 2² * 3³ = 2² * 3³ = 4 * 27 = 108
  • Let's solve 2² * 3³ step by step!
    • First, we know 2² means 2 × 2, which equals 4.
    • Next, 3³ means 3 × 3 × 3, which equals 27.
  • Now, multiply the results:
  • 4 * 27 = 108
    • So, 2² * 3³ = 108!

Multiplying 2² and 3³ with different bases requires calculating each exponent separately before multiplying the results together to get the final answer of 108.

Multiplying Exponents with Variables

The same rules apply when multiplying exponents with variables.

• Example with same base: When you multiply powers with the same base, just add the exponents!
  • For example:
  • a⁵ * a³ = a⁽⁵⁺³⁾ = a⁸
  • This means you add 5 + 3 to get 8.

• Example with Different Bases: If the bases are different, you can't combine them.
  • For example:
  • x³ * y²
  • You can't simplify this any further because the bases (x and y) are different.

When the bases are the same, you multiply exponents by adding the powers together. But if the bases are different, you can't simplify the expression any further.

Examples of Multiply and Divide Exponents

1. Simplify:  (5m³n²)² * (m⁴n)

• Distribute the exponent outside the first parentheses to each factor inside.

(5m³n²)² * (m⁴n)  = 5² * (m³)² * (n²)² * (m⁴n)

• Calculate the powers of the numerical and variable factors.

25 * m⁶ * n⁴ * m⁴n

• Since the bases'm' and 'n' are the same, add the exponents when multiplying.

25 * m⁽⁶⁺⁴⁾ * n⁽⁴⁺¹⁾

• Combine the terms to get the simplified expression: 25m¹⁰n⁵

2. Simplify: p⁸q⁶ / p³q⁴

• We have two different bases, 'p' and 'q'.
• Since the bases are the same for each variable, subtract the exponents in the denominator from the exponents in the numerator.
• p⁽⁸⁻³⁾ * q⁽⁶⁻⁴⁾
• Simplify the exponents:  p⁵q²