High Impact Tutoring Built By Math Experts
Personalized standards-aligned one-on-one math tutoring for schools and districts
Request a demo
In order to access this I need to be confident with:
Exponents Simplifying expressions Order of operations
Introduction
What is multiplying exponents?
↓
Multiplying exponents when the bases are the sameMultiplying exponents when the bases are different
Common Core State Standards
How to multiply exponents with the same base
Multiplying exponents examples
↓
Example 1: exponents with the same base, different exponents
Example 2: bases have a coefficient greater than 1.Example 3: with negative exponents
How to multiply exponents when the bases are different
↓
Example 4: with positive exponentsExample 5: with negative exponentsExample 6: with three terms
Teaching tips for multiplying exponents
Easy mistakes to make
Related laws of exponents lessons
Practice multiplying exponents questions
Multiplying exponents FAQs
Next lessons
Still stuck?
Math resources Algebra Laws of exponents
Multiplying exponents
Here you will learn about multiplying exponents, including how to apply the properties of exponents when multiplying exponents with the same base and with different bases.
Students will first learn about multiplying exponents as a part of expressions and equations in 8 th grade math and will expand on that knowledge into high school.
What is multiplying exponents?
Multiplying exponents is where you multiply terms that involve exponents, indices or powers. You can multiply exponents in various forms, including whole numbers, negative numbers, fractions, and decimals. There are two methods to multiply terms involving exponents,
- When the bases are the same, add the exponents.
For example,
a^{3} \times a^{4}=a^{3 \, + \, 4}=a^{7}
These questions usually ask to ‘simplify’ the calculation.
- When the bases are different, multiply the coefficients and keep the exponent the same.
For example,
2^{3} \times 3^{2}=8 \times 9=72
These questions usually ask to ‘evaluate’ the calculation.
What is multiplying exponents?
Multiplying exponents when the bases are the same
In order to multiply exponents when the bases are the same, you can use one of the laws of exponents.
- When multiplying exponents with the same base, add the powers.
a^{m} \times a^{n}=a^{m \, + \, n}
For example,
To simplify the following expression,
8^{3} \times 8^{4}
Write out each power in its expanded form.
\begin{aligned} {\color{red} 8^{3}} \times {\color{blue} 8^{4}} &= {\color{red}8 \times 8 \times 8} \times {\color{blue} 8 \times 8 \times 8 \times 8} \\\\ &=8^{7}\end{aligned}
Is there a quicker way to work this out?
Simplifying
8^{3} \times 8^{4} to 8^{3 \, + \, 4}
work out the simplified answer as,
8^{7}
8^{3} \times 8^{4}=8^{3 \, + \, 4}=8^{7}
The base has stayed the same and you have added the exponents together.
This is the multiplication law of exponents.
Multiplying exponents when the bases are different
- When multiplying exponents with different bases, multiply the coefficients and keep the exponent the same.
a^n \times b^n=(a \times b)^n
For example,
To evaluate the following expression:
2^{3} \times 3^{2}
Write each term of the calculation without using exponential notation. The exponent will tell you the number of times to represent the term.
2^{3} \times 3^{2}=2 \times 2 \times 2 \times 3 \times 3
\begin{aligned} 2^{3}&=2 \times 2 \times 2=8 \\\\ 3^{2}&=3 \times 3=9\end{aligned}
Work out the final answer by multiplying these numbers together.
\begin{aligned}2^{3} \times 3^{2} &=8 \times 9 \\\\ &=72\end{aligned}
Common Core State Standards
How does this relate to 8 th grade math?
- Grade 8: Expressions and Equations (8.EE.A.1)
Know and apply the properties of integer exponents to generate equivalent numerical expressions.
For example, 32 \times 3-5=3-3=\cfrac{1}{33}=\cfrac{1}{27} \, .
[FREE] Algebra Check for Understanding Quiz (Grade 6 to 8)
Use this quiz to check your grade 6 – grade 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th to 8th grade algebra topics to identify areas of strength and support!
DOWNLOAD FREE
[FREE] Algebra Check for Understanding Quiz (Grade 6 to 8)
Use this quiz to check your grade 6 – grade 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th to 8th grade algebra topics to identify areas of strength and support!
DOWNLOAD FREE
How to multiply exponents with the same base
In order to multiply exponents with different bases:
- Add the exponents.
- Multiply any coefficients.
Multiplying exponents examples
Example 1: exponents with the same base, different exponents
Simplify and leave your answer in exponential form.
a^{5} \times a^{2}
- Add the exponents.
5+2=7
2Multiply any coefficients.
The coefficients are both 1.
The answer is a^{7}.
Example 2: bases have a coefficient greater than 1 .
Simplify and leave your answer in index form.
4 a^{3} \times 7 a^{3}
Add the exponents.
Add together the exponents 3 and 3.
3+3=6
Multiply any coefficients.
Multiply 4 and 7 together.
4 \times 7 =28
The answer is 28a^{6}.
Example 3: with negative exponents
Simplify and leave your answer in index form.
7 a^{-6} \times 9 a^{-2}
Add the exponents.
Add together the exponents -6 and -2.
-6+-2=-6-2=-8
Multiply any coefficients.
Multiply the 7 and 9 together.
7 \times 9=63
=63a^{-8}
How to multiply exponents when the bases are different
In order to multiply exponents with different bases:
- Write out each term without the exponents.
- Work out the calculation.
Example 4: with positive exponents
Evaluate:
4^{2} \times 2^{4}
Write out each term without the exponents.
4^{2} \times 2^{4}=4 \times 4 \times 2 \times 2 \times 2 \times 2
Work out the calculation.
\begin{aligned} 4^{2}&=4 \times 4=16 \\\\ 2^{4}&=2 \times 2 \times 2 \times 2=16 \end{aligned}
So,
\begin{aligned}4^{2} \times 2^{4} &=16 \times 16 \\\\ &=256\end{aligned}
Example 5: with negative exponents
Evaluate:
3^4 \times 4^{-2}
Write out each term without the exponents.
When multiplying with a negative exponent, like 4^{-2}, you will find the reciprocal of the term.
3^4 \times 4^{-2}=3 \times 3 \times 3 \times 3 \times \cfrac{1}{4} \times \cfrac{1}{4}
Work out the calculation.
\begin{aligned}& 3^4=3 \times 3 \times 3 \times 3=81 \\\\ & 4^{-2}=\cfrac{1}{4} \times \cfrac{1}{4}=\cfrac{1}{16}\end{aligned}
So,
\begin{aligned}& 3^{4} \times 4^{-2}=81 \times \cfrac{1}{16} \\\\ & 3^{4} \times 4^{-2}=5.0625\end{aligned}
Example 6: with three terms
Evaluate:
3^{2} \times 2^{3} \times 10^{2}
Write out each term without the exponents.
3^{2} \times 2^{3} \times 10^{2}=3 \times 3 \times 2 \times 2 \times 2 \times 10 \times 10
Work out the calculation.
\begin{aligned} 3^{2}&=3 \times 3=9 \\\\ 2^{3}&=2 \times 2 \times 2=8 \\\\ 10^{2}&=10 \times 10=100 \end{aligned}
So,
\begin{aligned} 3^{2} \times 2^{3} \times 10^{2} &=9 \times 8 \times 100 \\\\ &=7200\end{aligned}
Teaching tips for multiplying exponents
- Before introducing multiplying exponents, students should be able to simplify expressions with the same base, understand the product rule, the power of a power rule, and the other exponent rules.
- Introduce students to real-life scenarios that use multiplying exponents, including scientific notation and compound interest.
- Provide worksheets with a range of different types of problems on various levels, including negative exponents, fractions as exponents, and mixed expressions.
Easy mistakes to make
- Using the multiplying exponents law with terms with different bases
For example,
You cannot simplify a^{4} \times b^{3} as the bases are different.
- Confusing exponents and fractional powers
Raising a term to the power of 2 means square it.
For example,
a^{2}=a \times aRaising a term to the power of \cfrac{1}{2} means find the square root of it.
For example,
a^{\frac{1}{2}}=\pm \sqrt{a}Raising a term to the power of 3 means cube it.
For example,
a^{3}=a \times a \times aRaising a term to the power of \cfrac{1}{3} means find the cube root of it.
For example,
a^{\frac{1}{3}}=\sqrt[3]{a}
Related laws of exponents lessons
- Dividing exponents
- Exponential notation
- Anything to the power of 0
- Negative exponents
- Cube root
- Square root
- Exponent rules
- Distributing exponents
Practice multiplying exponents questions
1. Simplify and leave your answer in index form.
x^{5} \times x^{2}
x^{10}
x^{7}
2x^{7}
10x
The product of the coefficients is 1, 1 \times 1=1.
The base numbers are the same, so add the exponents.
5+2=7
The answer is x^{7}.
2. Simplify and leave your answer in index form.
4 b^{3} \times b^{-7}
4b^{-4}
4b^{-21}
4b^{4}
4b^{-10}
The product of the coefficients is 4, 4 \times 1=4.
Because the same base number is the same, add the exponents.
3+(-7)=-4
The answer is 4b^{-4}.
3. Simplify and leave your answer in index form.
6 a^{5} \times 4 a^{-3}
10a^{2}
24a^{-15}
10a^{-15}
24a^{2}
The product of the coefficients is 24, 6 \times 4=24.
Because the base is the same number, add the index numbers.
5+(-3)=2
The correct answer is 24a^{2}.
4. Evaluate
3^{3} \times 2^{2}
6^{6}
6^{5}
108
36
As the base numbers are different, evaluate each term and then multiply,
3^{3} \times 2^{2}=27 \times 4=108
5. Evaluate
8^{2} \times 4^{-2}
4
32^{0}
32^{-4}
1,024
As the base numbers are different, evaluate each term and then multiply,
8^{2} \times 4^{-2}=64 \times \cfrac{1}{16}=4
6. Evaluate
2^{4} \times 5^{2} \times 3^{3}
30^{9}
30^{24}
10,800
720
As the base numbers are different, evaluate each term and then multiply,
2^{4} \times 5^{2} \times 3^{3}=16 \times 25 \times 27 = 10,800
Multiplying exponents FAQs
What is an exponent?
An exponent is a small number that is written above and to the right of a number, known as the base number. This indicates how many times a number is multiplied by itself (repeated multiplication). For example, 2^{3}, read as 2 to the third power, 2 is the base number and 3 is the exponent.
Can you multiply exponents that are fractions or decimals?
Yes, you can still apply the rules of exponents when dealing with exponential expressions that involve decimal or fractional exponents. For example, a^{\frac{1}{3}} \times a^{\frac{1}{3}}=a(^{\frac{1}{3}}+^{\frac{1}{3}})=a^{\frac{2}{3}}. You would find a common denominator, if needed, and then add the numerators similarly to adding fractions.
What is the power of a power property?
To find a power of a power, you will multiply powers, or multiply the exponents. For example, \left(x^n\right)^m=x^{n \cdot m}.
What is the rule for dividing with exponents?
When dividing exponents with the same base, subtract the powers to calculate the quotient. For example, \cfrac{x^n}{x^m}=x^{n-m}.
The next lessons are
- Scientific notation
- Math formulas
- Radicals
- Factoring
- Quadratic graphs
Still stuck?
At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.
Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.
Find out how we can help your students achieve success with our math tutoring programs.
Introduction
What is multiplying exponents?
↓
Multiplying exponents when the bases are the sameMultiplying exponents when the bases are different
Common Core State Standards
How to multiply exponents with the same base
Multiplying exponents examples
↓
Example 1: exponents with the same base, different exponents
Example 2: bases have a coefficient greater than 1.Example 3: with negative exponents
How to multiply exponents when the bases are different
↓
Example 4: with positive exponentsExample 5: with negative exponentsExample 6: with three terms
Teaching tips for multiplying exponents
Easy mistakes to make
Related laws of exponents lessons
Practice multiplying exponents questions
Multiplying exponents FAQs
Next lessons
Still stuck?
x
[FREE] Common Core Practice Tests (Grades 3 to 6)
Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.
40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!
Download free
We use essential and non-essential cookies to improve the experience on our website. Please read our Cookies Policy for information on how we use cookies and how to manage or change your cookie settings.Accept
