Multiplying and Dividing exponents

Multiplying and Dividing exponents
Go back to  'Exponents Squares Cubes'

When we multiply two exponential terms with the same base, for example, \({2^3} \times {2^4}\), the first exponential term gives 3 twos, while the second gives 4 twos. So we can say that

\[{2^3} \times {2^4} = {2^{\left( {3 + 4} \right)}} = {2^7}\]

Carefully observe how the exponents have added. This is a general behavior of exponential terms: while multiplying exponents with same base, the exponents add. In a sense, the contributions of the base from both the terms are adding up. It is easy to see that this property will hold for the multiplication of any number of exponential terms with the same base:

multiplication of polynomial

Now, suppose that we divide two exponential terms that have the same base: \({3^7} \div {3^4}\). The first exponential term has 7 threes, while the second has 4 threes. Dividing the first by the second will take away 4 threes from the first term. Alternatively, you can say that you have to contribute 3 more threes to the second term to be able to generate the first term. Thus,

\[{3^7} \div {3^4} = {3^{\left( {7 - 4} \right)}} = {3^3}\]

So, whenever we divide two exponential terms with the same base, the exponents subtract. In a sense, the divisor takes away some bases from the dividend.

In this mini-lesson, we are going to learn about multiplying and dividing numbers with exponents by understanding the rules for multiplying exponents, rules for dividing exponents. We will learn how to apply them while solving problems. We will also discover interesting facts around them.

Lesson Plan

How to Multiply And Divide Exponents With the Same Base?

Multiplying Exponents With the Same Base

Before understanding how to multiply exponents with the same base, let's see some examples of exponents with the same base.

The exponents with the same base are

exponent and base

When multiplying exponents with same base we will be going to use the identity or it sometimes referred to as rules for multiplying exponents.

exponent identity

\[\text{Where a is the same base and m and n are the exponents}\]

Let's take an example, multiply \(3^3\) and \(3^4\)

applying the identity \(a^m \times a^n = a^{m + n}\)

\[3^3 \times 3^4 = 3^{3 + 4} = 3^7 = 2187\]

Dividing Exponents With the Same Base

To divide the exponents with the same base, we are going to use the identity or it sometimes referred to as rules for dividing exponents.. 

exponent identity

\[\text{Where a is the same base and m and n are the exponents}\]

Let's take an example, divide \(7^6\) by \(7^2\)

applying the identity \(a^m \div a^n = a^{m - n}\)

\[7^6 \div 7^2 = 7^{6 - 2} = 7^4 = 2401\]


How to Multiply And Divide Exponents With Different Bases?

Multiplication With Different Base And Same Exponent

When multiplying exponents with different bases and the same exponent, we are going to use identity. 

exponent identity

\[\text{Where a and b are the different bases and m is the exponent}\]

Let's take an example. Multiply \(11^4\) and \(3^4\)

Applying the identity \(a^m \times b^m = {\left( ab \right)}^{m}\) 

\[11^4 \times 3^4 = {(11 \times 3)}^{4} = 333^4 = 12296370321\]

Multiplication With Different Base And Different Exponent

When multiplying exponents with different bases and different exponents we do not use any identity.

Let's take an example, multiply \(9^2\) and \(4^3\)

\[9^2 \times 4^3 = {(81 \times 64)} = 5184\]

Division With Different Base And Same Exponent

To divide the exponents with the different bases and the same exponent, we are going to use identity. 

exponent identity

\[\text{Where a and b are the different bases and m is the exponent}\]

Let's take an example. Divide \(12^3\) by \(3^3\)

Applying the identity \(a^m \div b^m = {\left( \frac{a}{b} \right)}^{m}\)

\[12^3 \div 3^3 = {\left( \frac{12}{3} \right)}^{3} = 4^3 = 64\]

Division With Different Base And Different Exponent

To divide the exponents with different bases and different exponents we do not use any identity.

Let's take an example, divide \(8^3\) by \(5^2\)

\[8^3 \div 5^2 = {(512 \div 25)} = 20.48\]

 
Thinking out of the box
Think Tank
  • What if we need to prove \(a^0 = 1\) without using logarithm and factorial? Think about it.

How to Multiply And Divide Fractional Exponents?

Multiplying Fractional Exponents With the Same Base

Before understanding how to multiply fractional exponents with the same base, let's see some examples of fractional exponents with the same base.

Some fractional exponents with the same base are

exponent and base

To multiply the fractional exponents with the same base, we are going to use identity. 

exponent identity

\[\text{Where a is the same base and m and n are the fractional exponents}\]

Let's take an example. Multiply \(2^{\left( \frac{1}{2} \right)}\) and \(2^{\left( \frac{3}{4} \right)}\)

applying the identity \(a^m \times a^n = a^{m + n}\)

\[2^{\left( \frac{1}{2} \right)} \times 2^{\left( \frac{3}{4} \right)} = 2^{\left( \frac{3}{4} + \frac{1}{2} \right)} = 2^{\left( \frac{5}{4} \right)} = 2.3784\]

Dividing Fractional Exponents With the Same Base

To divide the fractional exponents with the same base, we are going to use identity.

exponent identity

\[\text{Where a is the same base and m and n are the fractional exponents}\]

Let's take an example, divide \(3^{\left( \frac{3}{2} \right)}\) by \(3^{\left( \frac{1}{2} \right)}\)

applying the identity \(a^m \div a^n = a^{m - n}\)

\[3^{\left( \frac{3}{2} \right)} \div 3^{\left( \frac{1}{2} \right)} = 3^{\left( \frac{3}{2} - \frac{1}{2} \right)} = 3^1 = 3\]


How to Multiply And Divide Exponents With Variables?

Variable as the Base

We can apply the same rule for the variable as the base that we were applying for numbers as the base.

Let's take an example for multiplication, multiply \(x^2\) and \({2x}^3\)

Applying the identity \({(ab)}^m = a^m \times b^m\)

\[x^2 \times {2x}^3 \\[0.2cm]
= x^2 \times x^3 \times 2^3 \\[0.2cm]
= x^{(2 + 3)} \times 8 \\[0.2cm]
= 8x^5\]

Let's take an example for division. Divide \(x^5\) by \(x^3\)

Applying the identity \(a^m \div a^n = a^{(m - n)}\)

\[x^5 \div x^3 = x^{(5 - 3)} = x^2\]

Variable as the Exponent

We can apply the same rule for variable as exponents that we were applying for numbers as the base.

Let's take an example. Multiply \(3^x\) and \(3^{(x + 1)}\)

Applying the identity \(a^m \times a^n = a^{(m + n)}\)

\[3^x \times 3^{(x + 1)} = 3^{x(x + 1)}\]

Let's take an example for division. Divide \(5^{(2x - 1)}\) by \(5^{(x + 1)}\)

Applying the identity \(a^m \div a^n = a^{(m - n)}\)

\[5^{(2x - 1)} \div 5^{(x + 1)} = 5^{(2x - 1 - (x + 1))} = 5^{(2x - 1 - x - 1)} = 5^{(x - 2)}\]

Explore the multiplying and dividing exponents calculator below to understand more about multiplication and division of exponents.

 
important notes to remember
Important Notes
  • If the bases of the exponents are equal in any equation then exponents must be equal.
    \[a^p = a^q \text{ then } p = q\]
  • If we multiply two exponents with the same base then the powers will add.
  • If we divide two exponents with the same base then the powers will subtract.

Solved Examples on Multiplying And Dividing Exponents

Example 1

 

 

Help Kevin to find the value of the given exponents problem \(3^2 \times 9^2\)

Solution

In the above problem, the bases are not the same. But we can make the base same.
\[\begin{align}
&= 3^2 \times 9^2 \\[0.2cm]
&= 3^2 \times {(3^2)}^2 \\[0.2cm]
&= 3^2 \times 3^4 \\[0.2cm]
&= 3^{(2 + 4)} \\[0.2cm] 
&= 3^6 \\[0.2cm]
&= 729 \end{align}\]

\(\therefore 729\) 
Example 2

 

 

Find the value of the expression \( 3{(a^4b^3)}^{10} \times 5{(a^2b^2)}^{3} \)

Solution

We will use the identity \({(m^h)}^k = m^{hk}\) and \(m^h \times m^k = m^{(h + k)}\)
Let the value of the expression be \(y\)
\[\begin{align}
y &= 3{(a^4b^3)}^{10} \times 5{(a^2b^2)}^{3} \\[0.2cm]
y &= (3 \times 5)(a^{(4 \times 10)}b^{(3 \times 10)}) \times (a^{(2 \times 3)}b^{(2 \times 3)}) \\[0.2cm]
y &= (15)(a^{(40)}b^{(30)}) \times (a^{(6)}b^{(6)}) \\[0.2cm]
y &= (15)(a^{(40)}a^{(6)}b^{(30)}b^{(6)}) \\[0.2cm]
y &= (15)(a^{(40 + 6)}b^{(30 + 6)}) \\[0.2cm]
y &= (15)(a^{(46)}b^{(36)}) \end{align}\]

\(\therefore 15(a^{(46)}b^{(36)})\) 
Example 3

 

 

What is the value of the expression \(\dfrac{a^{(n - 2)}}{a^{(2n - 4)}}\) equal to? 

Solution

Let the value of the expression be \(y\) 

Applying the identity: \(\dfrac{m^h}{m^k} = m^{(h - k)}\)
\[\begin{align}
y &= \dfrac{a^{(n - 2)}}{a^{(2n - 4)}} \\[0.2cm]
y &= a^{(n - 2 - (2n - 4))} \\[0.2cm]
y &= a^{(n - 2 - 2n + 4)} \\[0.2cm]
y &= a^{(-n + 2)} \\[0.2cm]
y &= a^{(2 - n)} \end{align}\]

\(\therefore a^{(2 - n)}\) 

Interactive Questions on Multiplying and Dividing Exponents

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

This mini-lesson introduced you to the fascinating concept of multiplying and dividing exponents. The math journey around multiplying and dividing exponents starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts are dedicated to making learning fun for our favorite readers, the students! Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. Be it simulations, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.


FAQs on Multiplying And Dividing Exponents

1. What are the rules for dividing exponents?

To divide the numbers or variables with the same base, we apply the identity \(a^m \div a^n = a^{(m - n)}\)

To divide the numbers or variables with different bases, we apply the identity \(a^m \div b^m = {\left( \dfrac{a}{b} \right)}^m\)

2. Do you multiply exponents when distributing?

We can distribute exponents on multiplication and division

For example, \(6^2 = {(2 \times 3)}^2 = 2^2 \times 3^2\)

And, \(5^3 = {\left( \dfrac{10}{2} \right)}^3 = {\left( \dfrac{10^3}{2^3} \right)}\)

3. How do you solve exponents in parentheses?

Exponents inside parentheses can be solved using the identity \({(a^m)}^n = a^{mn}\).

For example, \({(4^2)}^3 = 4^{2 \times 3} = 4^6 = 4096\)

4. Can you multiply exponents with different coefficients?

Yes, we can multiply exponents with different coefficients.

For example, \(3^{2x} \times 3^{3x} = 3^{2x + 3x} = 3^{5x}\)

5. When multiplying, do you add exponents?

Yes. While multiplying, we add the exponents if the bases are the same.

6. How do you multiply exponents inside parentheses?

Exponents inside a parenthesis can be solved using the identity \({(a^m)}^n = a^{mn}\).

For example, \({(3^3)}^5 = 3^{3 \times 5} = 3^15 = 14348907\)

7. When dividing exponents, do you divide powers?

While performing division, we subtract the powers.

For example \(9^3 \div 9^2 = 9^{3 - 2} = 9^1 = 9\)

8. How do you divide numbers using a logarithm?

We can divide the numbers with a logarithm.

For example, \(log_2 10 \div log_2 20 = log_10 20\)

9. Can we distribute exponents over division?

Yes, we can distribute exponents over division.

For example, \({\left( \dfrac{7}{2} \right)}^3 = \dfrac{7^3}{2^3} = \dfrac{343}{8}\)

10. How to multiplying and dividing negative exponents?

When multiplying and dividing negative exponents we perform the calculations shown below.

For example, \(2^{-3} \div 2^{-4} = \dfrac{2^{-3}}{2^{-4}} = 2^{(-3 - (-4))} = 2^{(-3 + 4)} = 2^1 = 2\)

Download Exponents and Logarithms Worksheets
Exponents and Logarithms
Grade 9 | Questions Set 1
Exponents and Logarithms
Grade 9 | Answers Set 1
Exponents and Logarithms
Grade 9 | Questions Set 2
Exponents and Logarithms
Grade 9 | Answers Set 2
  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus