how to multiply square roots

Multiplying Square Roots: 3 Easy Methods [with Examples]

Your students know how to multiply exponents, but now it’s time to teach them about multiplying square roots and the wonderful world of pre-algebra. But, you fear they might be thinking: “In math class, we learned more about algebra, such as X + 10 = Y, but why should I care?”

You want them to realize what French mathematician, Jean de Rond d’Alembert, famously said: “algebra is generous; she often gives more than is asked of her.”

Just because you don’t see Xs and Ys doesn’t mean you aren’t using algebra every day. Learning how to multiply square roots is one cobblestone on the scenic road to understanding algebra’s real-life relevance.

Well, another day has passed. I used Algebra so many times that I didn't even realize I was doing it. (There. Fixed that for you.)

Educators like yourself know it’s not always easy to make these abstract and complex concepts engaging and fun.

This blog post — divided into three parts — aims to change that!

  • Understanding square roots
  • How to multiply square roots
  • Engaging ways to enforce students’ square root knowledge

Part One: What is a “square root”?

The square root of a number refers to the factor you can multiply by itself to get that number. Put another way, finding the square root is the opposite process of squaring a number.

It’s only possible to take the square root of non-negative numbers — even ones that don’t yield whole numbers. This is because any number times itself is positive, or zero — you can never get a negative product when you square a negative number. Like you saw above, square roots undo squaring, so it’s impossible for negative numbers to have square roots.

That said, perfect square numbers are the most effective ones to use when teaching students about multiplying square roots.

list of perfect square roots

In the visual below we see the square root of 16 is 4 because 4 squared, or 42, is 16.

rules for multiplying square roots

In math class, this kind of equation would look like:

the answer to square root of sixteen is four

Students seeing this for the first time will likely have some questions: What’s that check mark-looking symbol? Why’s there a tiny number over it?

Brett Berry, the founder of Math Hacks, created a clear and concise image with all the root terminology in her article on understanding logarithms and roots.

Root terminology includes radical symbol, solution or root, root index, and radicand.

Many questions about finding square roots don’t include the root index. However, root indexes are necessary when calculating higher indexed roots like cube, fourth, or fifth roots.

When will I ever need to multiply square roots?

No matter how well you teach these algebraic concepts, students will always ask that question. ☝️

And you should be prepared to provide them with legitimate answers. For example, multiplying square roots can be important for:

  • Knowing the square footage of their future homes
  • Architects
  • Artists
  • Carpenters
  • Construction workers
  • Designers
  • Engineers

There are more, too! While some will have to calculate equations every day, others will use the concepts to make estimates. One thing’s for sure — people in those professions learned math in school as kids and they’re still using it today!

Part Two: 3 Simple methods for multiplying square roots

Multiplying square roots without coefficients

1. Multiply each radicand the same way you would without the radical, or square root symbol.

multiplying 2 square roots

2. Simplify the radicand by factoring out all perfect squares. If there are no perfect squares in the radicand, it’s already simplified. In this case, you can simplify √98 to √2 and √49 — a perfect square.

multiplying by square roots

3. Take out the square root of the perfect square. In this example, simplify √49 to 7 and place it in front of the remaining expression, √2.

multiplying different square roots

Multiplying square roots with coefficients

1. Multiply coefficients in front of radical signs, if any.

multiplying square roots with whole numbers

2. Multiply each radicand the same way you would without the radical, or square root symbol.

multiplying square roots together

3. Simplify the radicand by factoring out all perfect squares. In this example, you can simplify √40 to √4 and √10.

multiply square roots with coefficients

4. Take out the square root of the perfect square and multiply it by the coefficient. In the example, simplify √4 to 2 and multiply it by 6.

square root multiplication simplified

Multiplying square roots with variables

In addition to numbers, radicals can contain other things like variables and exponents. Simplifying radicals with variables follows the same rules as one that only has numbers.

1. Multiply the radicands. If there are coefficients multiply them together as well.

multiplying square roots with x

Note: In order to multiply radicals containing variables, the:

  • Root index must be the same
  • Value of x — along with any other variables — must be greater than or equal to zero

2. Find the prime factorizations to identify the perfect square factors. To do this, you can use a factor tree — like in the image below.

factor tree example

For example, 2 and 9 equal 18, and 9 simplifies further to 3 and 3. So, the prime factorizations and perfect square factors for 18 would be 2, 3, and 3. The prime factorization of 30 is 2, 3, and 5. You can also break down and rearrange the variable exponents — you can rewrite x3 as x2 and x.

multiplying two square roots with variables

3. Simplify the radicand. Before simplifying, it’s helpful to identify pairs of equal factors because they’re examples of perfect squares. Because squaring and finding the square root are inverse operations, they cancel each other out. For example, √22 simplifies to 2.

simplifying square roots with variables

Part Three: Activities to enforce students’ square root knowledge

Create radical number towers

Create radical number towers to help teach students about square roots.
Source: Focus on Math

After teaching your students about square roots — and how to multiply them — split them into groups to make some radical number towers! Starting with √1, have them make a tower going all the way up to the square root of your choice.

On or beside each part of the tower, you can have students write both the square and square root. This activity is an effective visual representation of what each one looks like and should help them remember the most common square roots.

Escape the radicals maze

multiplying square roots with radicals
Source: Lisa Tarman

Lisa Tarman, an educator from Pennsylvania, has created hundreds of teaching resources. Her “Simplifying Radicals Maze” is a refreshing and engaging take on traditional worksheets.

Instruct your students to start in the top left corner. They’ll have to simplify radicals in order to make their way to the end of the maze. Access Tarman’s free fun worksheet and answer key here.

If you want a multiplying radicals maze, check out this one from Teachers Pay Teachers!

Play Prodigy

Example of what a Prodigy Math battle looks like.

Prodigy is a no-cost, adaptive math game used by one and a half million teachers and more than 50 million students around the world! It offers content from every major math topic and covers 1st to 8th grade, including how to:

  • Evaluate perfect roots
  • Rewrite exponents as roots

Example of a square roots question in Prodigy Math

Using Prodigy in your classroom will help students build math fluency and confidence in future high school and college-level math courses. Your class will explore a world filled with exciting quests that deliver personalized, curriculum-aligned content and real-time student data.

With these methods and activities in mind, you can see how multiplying square roots doesn’t have to remain irrelevant or intimidating for you or your students.

When used effectively, activities like the ones above can help solidify students’ understanding and create levels of engagement teachers seldom witness in math class.

Are you an educator? Customize math questions to supplement class material and differentiate instruction, addressing each student’s trouble spots.

Prodigy also features powerful and immediate reporting tools for both teachers and parents. From Progress Reports to Usage Reports and more, use your student or child’s data to identify where they’re excelling or struggling so you can adjust the content in game for them.

Click here or the banner below to get start in less than five minutes!

Curriculum-aligned math has never been so fun, sign up now!

Jordan Nisbet

Jordan crafts content for Prodigy — and wishes the game existed when he was in school. He's interested in education and passionate about helping build up the next generation!

Leave a Reply

Your email address will not be published. Required fields are marked *