How to Multiply Square Roots

Co-authored by David Jia

Last Updated: May 27, 2024 Fact Checked

You can multiply square roots, a type of radical expression, just as you might multiply whole numbers. Sometimes square roots have coefficients (an integer in front of the radical sign), but this only adds a step to the multiplication and does not change the process. The trickiest part of multiplying square roots is simplifying the expression to reach your final answer, but even this step is easy if you know your perfect squares.

Method 1

Method 1 of 2:

Multiplying Square Roots Without Coefficients

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b0\/Multiply-Square-Roots-Step-1-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-1-Version-3.jpg","bigUrl":"\/images\/thumb\/b\/b0\/Multiply-Square-Roots-Step-1-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-1-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

1
Multiply the radicands. A radicand is a number underneath the radical sign. ^{[1]
X
Research source} To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign. ^{[2]
X
Research source}
- For example, if you are calculating ${\sqrt {15}}\times {\sqrt {5}}$ , you would calculate $15\times 5=75$ . So, ${\sqrt {15}}\times {\sqrt {5}}={\sqrt {75}}$ .
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/9c\/Multiply-Square-Roots-Step-2-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-2-Version-3.jpg","bigUrl":"\/images\/thumb\/9\/9c\/Multiply-Square-Roots-Step-2-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-2-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

2
Factor out any perfect squares in the radicand. To do this, see whether any perfect square is a factor of the radicand. ^{[3]
X
Research source} If you cannot factor out a perfect square, your answer is already simplified and you need not do anything further.
- A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself. ^{[4]
  X
  Research source} For example, 25 is a perfect square, because $5\times 5=25$ .
- For example, ${\sqrt {75}}$ can be factored to pull out the perfect square 25:
  ${\sqrt {75}}$
  = ${\sqrt {25\times 3}}$
Advertisement
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/1\/16\/Multiply-Square-Roots-Step-3-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-3-Version-3.jpg","bigUrl":"\/images\/thumb\/1\/16\/Multiply-Square-Roots-Step-3-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-3-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

3
Place the square root of the perfect square in front of the radical sign. Keep the other factor under the radical sign. This will give you your simplified expression.
- For example, ${\sqrt {75}}$ can be factored as ${\sqrt {25\times 3}}$ , so you would pull out the square root of 25 (which is 5):
  ${\sqrt {75}}$
  = ${\sqrt {25\times 3}}$
  = $5{\sqrt {3}}$
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/0\/06\/Multiply-Square-Roots-Step-4-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-4-Version-3.jpg","bigUrl":"\/images\/thumb\/0\/06\/Multiply-Square-Roots-Step-4-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-4-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

4
Square a square root. In some instances, you will need to multiply a square root by itself. Squaring a number and taking the square root of a number are opposite operations; thus, they undo each other. The result of squaring a square root, then, is simply the number under the radical sign. ^{[5]
X
Research source}
- For example, ${\sqrt {25}}\times {\sqrt {25}}=25$ . You get that result because ${\sqrt {25}}\times {\sqrt {25}}=5\times 5=25$ .

Method 2

Method 2 of 2:

Multiplying Square Roots With Coefficients

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5b\/Multiply-Square-Roots-Step-5-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-5-Version-3.jpg","bigUrl":"\/images\/thumb\/5\/5b\/Multiply-Square-Roots-Step-5-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-5-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

1
Multiply the coefficients. A coefficient is a number in front of the radical sign. To do this, just ignore the radical sign and radicand, and multiply the two whole numbers. Place their product in front of the first radical sign.
- Pay attention to positive and negative signs when multiplying coefficients. Don't forget that a negative times a positive is a negative, and a negative times a negative is a positive.
- For example, if you are calculating $3{\sqrt {2}}\times 2{\sqrt {6}}$ , you would first calculate $3\times 2=6$ . So now your problem is $6{\sqrt {2}}\times {\sqrt {6}}$ .
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/e8\/Multiply-Square-Roots-Step-6-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-6-Version-3.jpg","bigUrl":"\/images\/thumb\/e\/e8\/Multiply-Square-Roots-Step-6-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-6-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

2
Multiply the radicands. To do this, multiply the numbers as if they were whole numbers. Make sure to keep the product under the radical sign. ^{[6]
X
Research source}
- For example, if the problem is now $6{\sqrt {2}}\times {\sqrt {6}}$ , to find the product of the radicands, you would calculate $2\times 6=12$ , so ${\sqrt {2}}\times {\sqrt {6}}={\sqrt {12}}$ . The problem now becomes $6{\sqrt {12}}$ .
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/e\/ec\/Multiply-Square-Roots-Step-7-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-7-Version-3.jpg","bigUrl":"\/images\/thumb\/e\/ec\/Multiply-Square-Roots-Step-7-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-7-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

3
Factor out any perfect squares in the radicand, if possible. You need to do this to simplify your answer. ^{[7]
X
Research source} If you cannot pull out a perfect square, your answer is already simplified and you can skip this step.
- A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself. ^{[8]
  X
  Research source} For example, 4 is a perfect square, because $2\times 2=4$ .
- For example, ${\sqrt {12}}$ can be factored to pull out the perfect square 4:
  ${\sqrt {12}}$
  = ${\sqrt {4\times 3}}$
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/98\/Multiply-Square-Roots-Step-8-Version-3.jpg\/v4-460px-Multiply-Square-Roots-Step-8-Version-3.jpg","bigUrl":"\/images\/thumb\/9\/98\/Multiply-Square-Roots-Step-8-Version-3.jpg\/aid326954-v4-728px-Multiply-Square-Roots-Step-8-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

4
Multiply the square root of the perfect square by the coefficient. Keep the other factor under the radicand. This will give you your simplified expression.
- For example, $6{\sqrt {12}}$ can be factored as $6{\sqrt {4\times 3}}$ , so you would pull out the square root of 4 (which is 2) and multiply it by 6:
  $6{\sqrt {12}}$
  = $6{\sqrt {4\times 3}}$
  = $6\times 2{\sqrt {3}}$
  = $12{\sqrt {3}}$

Calculator, Practice Problems, and Answers

Sample Multiplying Square Roots Calculator

Sample Multiplying Square Roots Practice Problems

Sample Multiplying Square Roots Practice Answers

Community Q&A

Search

Add New Question

Question

We are not allowed to use a calculator, so how do I multiply a whole number by a square root?

Community Answer

When you multiply a whole number by a square root, you just put the two together, with the whole number in front of the square root. For example, 2 * (square root of 3) = 2(square root of 3). If the square root has a whole number in front of it, multiply the whole numbers together. So 2 * 4(square root of 3) = 8(square root of 3).

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 67 Helpful 98
Question

What is 2 root 3 times root 3?

Donagan

Top Answerer

√3 times √3 equals 3. Two times that is 6.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 79 Helpful 97
Question

What is 4 divided by square root of 5?

Community Answer

(4√5)/5. Since radicals are not supposed to be in the denominator, you multiply by √5/√5 to get (4√5)/5.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 68 Helpful 87

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Video

Tips

Follow the usual sign rules to determine whether the new coefficient should be positive or negative. A positive coefficient multiplied by a negative coefficient will be negative. Two positive coefficients multiplied together or two negative coefficients multiplied together will be positive.

Thanks
Helpful 0 Not Helpful 1
Always remember your perfect squares because it will make the process much easier!

Thanks
Helpful 0 Not Helpful 0
All terms under the radicand are always positive, so you will not have to worry about sign rules when multiplying radicands.

Thanks
Helpful 0 Not Helpful 0

Submit a Tip

All tip submissions are carefully reviewed before being published

Name

Please provide your name and last initial

Submit

Thanks for submitting a tip for review!

Things You'll Need

Pencil
Paper
Calculator

You Might Also Like

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

Log in

How to Multiply Square Roots

Steps

Multiplying Square Roots Without Coefficients

Multiplying Square Roots With Coefficients

Calculator, Practice Problems, and Answers

Community Q&A

Video

Tips

Things You'll Need

You Might Also Like

References

About This Article

Reader Success Stories

Did this article help you?

Quizzes

You Might Also Like

Featured Articles

Trending Articles

Featured Articles

Featured Articles

Watch Articles

Trending Articles

Quizzes

Explore Categories