How to Multiply Exponents

Co-authored by David Jia

Last Updated: October 21, 2022

Exponents are a way to identify numbers that are being multiplied by themselves. They are often called powers. You will come across exponents frequently in algebra, so it is helpful to know how to work with these types of expressions. You can multiply exponential expressions just as you can multiply other numbers. If the exponents have the same base, you can use a shortcut to simplify and calculate; otherwise, multiplying exponential expressions is still a simple operation.

Method 1

Method 1 of 3:

Multiplying Exponents with the Same Base

1
Make sure the exponents have the same base. The base is the large number in the exponential expression. You can only use this method if the expressions you are multiplying have the same base.
- For example, you can use this method to multiply $5^{2}\times 5^{3}$ , because they both have the same base (5). On the other hand, you cannot use this method to multiply $5^{2}\times 2^{3}$ , because they have different bases (5 and 2).
2
Add the exponents together. Rewrite the expression, keeping the same base but putting the sum of the original exponents as the new exponent.^{[1]
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- For example, if you are multiplying $5^{2}\times 5^{3}$ , you would keep the base of 5, and add the exponents together:
  $5^{2}\times 5^{3}$
  $=5^{2+3}$
  $=5^{5}$
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3
Calculate the expression. An exponent tells you how many times to multiply a number by itself.^{[2]
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Research source} You can use a calculator to easily calculate an exponential expression, but you can also calculate by hand.
- For example $5^{5}=5\times 5\times 5\times 5\times 5$
  $5^{5}=3,125$
  So, $5^{2}\times 5^{3}=3,125$

Method 2

Method 2 of 3:

Multiplying Exponents with Different Bases

1
Calculate the first exponential expression. Since the exponents have different bases, there is no shortcut for multiplying them. Calculate the exponent using a calculator or by hand. Remember, an exponent tells you how many times to multiply a number by itself.
- For example, if you are multiplying $2^{3}\times 4^{5}$ , you should note that they do not have the same base. So, you will first calculate $2^{3}=2\times 2\times 2=8$ .
2
Calculate the second exponential expression. Do this by multiplying the base number by itself however many times the exponent says.
- For example, $4^{5}=4\times 4\times 4\times 4\times 4=1024$
3
Rewrite the problem using the new calculations. Following the same example, your new problem becomes $8\times 1024$ .
4
Multiply the two numbers. This will give you the final answer to the problem.
- For example: $8\times 1024=8192.$ So, $2^{3}\times 4^{5}=8,192$ .

Method 3

Method 3 of 3:

Multiplying Mixed Variables with Exponents

1
Multiply the coefficients. Multiply these as you would any whole numbers. Move the number to the outside of the parentheses.
- For example, if multiplying $(2x^{3}y^{5})(8xy^{4})$ , you would first calculate $((2)x^{3}y^{5})((8)xy^{4})=16(x^{3}y^{5})(xy^{4})$ .
2
Add the exponents of the first variable. Make sure you are only adding the exponents of terms with the same base (variable). Don’t forget that if a variable shows no exponent, it is understood to have an exponent of 1.^{[3]
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- For example:
  $16(x^{3}y^{5})(xy^{4})=16(x^{3})y^{5}(x)y^{4}=16(x^{3+1})y^{5}y^{4}=16(x^{4})y^{5}y^{4}$
3
Add the exponents of the remaining variables. Take care to add exponents with the same base, and don’t forget that variables with no exponents have an understood exponent of 1.
- For example:
  $16(x^{4})y^{5}y^{4}=16x^{4}y^{5+4}=16x^{4}y^{9}$

Community Q&A

Question

What is the solution for 3.5 x 10 to the fourth power?

Community Answer

10^4 = 10 x 10 x 10 x 10 = 10,000, so you are really multiplying 3.5 x 10,000. The shortcut is that, when 10 is raised to a certain power, the exponent tells you how many zeros. 10^4 = 1 followed by 4 zeros = 10,000. Thus, you can just move the decimal point to the right 4 spaces: 3.5 x 10^4 = 35,000.
Question

How do I divide exponents that don't have the same base?

Community Answer

To learn how to divide exponents, you can read the following article: http://www.wikihow.com/Divide-Exponents
Question

How do I write 0.0321 in scientific notation?

Donagan

Top Answerer

0.0321 = 3.21 x 10^(-2).