Multiplying Radicals - Math Steps, Examples & Questions

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Multiplying radicals

Here you will learn all about how to multiply square root expressions (any root expression) and when to simplify the product.

Students first learn how to multiply radicals in algebra and build upon those strategies as they progress through high school math.

What is multiplying radicals?

Multiplying radicals is where you can multiply two or more radical expressions together. Unlike adding radicals, the number under the radical sign does not have to be the same in order to multiply.

When multiplying radical expressions, multiply the coefficients together, the radicands (number under the root symbol) together, and then simplify the expression if necessary.

The strategy of multiplying radical expressions is similar to multiplying algebraic expressions.

Recall multiplying $2a\cdot{3a}.$ Multiply the coefficients together and the variables together (remember to add exponents when multiplying).

2a\cdot{3a}=6 a^2

To multiply $4a\cdot{9},$ multiply the coefficients together. Since only one expression has a variable, remember to bring that down into the answer.

4a\cdot{9}=36a

This strategy can be applied to radical expressions.

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Let’s multiply $(2\sqrt{3})(3\sqrt{5}).$ Multiply the whole number coefficients together and the numbers under the radical symbol or square root symbol.

(2\sqrt{3})(3\sqrt{5})=6\sqrt{15}

The number under the radical, $15,$ cannot be simplified as the largest square number that is a factor of $15$ is $1,$ so leave the answer as is.

Now let’s multiply $(7 \sqrt{2})(8).$ Multiply the whole number coefficients together. Since only one expression has a radical, be sure to include that in the answer.

(7\sqrt{2})(8)=56\sqrt{2}

The number under the radical, $2,$ cannot be simplified so leave the answer as is.

You can also multiply radical expression binomials using the distributive property, similarly to the way you multiply algebraic expressions.

For example, let’s multiply $(2\sqrt{5}-4)(3\sqrt{2}+9)$

Using the distributive property, you find the product:

(2\sqrt{5}-4)(3\sqrt{2}+9)=6\sqrt{10}+18\sqrt{5}-12\sqrt{2}-36

Let’s look at when a radical expression is squared, like in the example of $(3 \sqrt{6}+4)^2$

When a number or expression is squared it means to multiply it to itself. So, in this case $(3 \sqrt{6}+4)^2$ means $(3 \sqrt{6}+4)(3 \sqrt{6}+4)$

Using the distributive property, multiply the expressions together,

\begin{aligned}(3\sqrt{6}+4)^{2}&=9\sqrt{36}+12\sqrt{6}+12\sqrt{6}+16 \\\\ &=9\times{6}+24\sqrt{6}+16 \\\\ &=70+24\sqrt{6} \end{aligned}

What is multiplying radicals?

Common Core State Standards

How does this relate to high school math?

High school- The Real Number System (HSN-RN.B.3)
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

How to multiply radicals

In order to multiply radicals:

Multiply the coefficients together and the radicands together.
Simplify the product.

Multiplying radicals examples

Example 1: simplify multiplication

Multiply the radical expressions.

2\sqrt{2}\times{5}\sqrt{3}

Multiply the coefficients together and the radicands together.

Multiply the coefficients together: $2\times{5}=10$

Multiply the radicands together: $\sqrt{2}\times\sqrt{3}=\sqrt{2\times3}=\sqrt{6}$

2Simplify the product.

The product of $2\sqrt{2}\times{5}\sqrt{3}=10\sqrt{6}$

$10 \sqrt{6}$ cannot be simplified as the greatest factor that is a square number is $1,$ so that is the most simplified answer.

Example 2: multiplication with simplifying

Multiply the radical expression.

5\sqrt{3}\times{2}\sqrt{6}

Multiply the coefficients together and the radicands together.

Multiply the coefficients together: $5\times{2}=10$

Multiply the radicands together: $\sqrt{3}\times\sqrt{6}=\sqrt{3\times6}=\sqrt{18}$

Simplify the product.

The product of $5\sqrt{3}\times{2}\sqrt{6}=10\sqrt{18}$

$10\sqrt{18}$ can be simplified further because $18$ has a perfect square factor.

$10\sqrt{18}=10\sqrt{9\times{2}}=10\sqrt{9}\times\sqrt{2}=10\times{3}\times\sqrt{2}=30\sqrt{2}$

The simplified product of $5\sqrt{3}\times{2}\sqrt{6}=30\sqrt{2}$

Example 3: multiply coefficient of 1

Multiply the radical expression $(12\sqrt{6})(\sqrt{8}).$

Multiply the coefficients together and the radicands together.

Multiply the coefficients together: $12\times{1}=12$

Multiply the radicands together: $\sqrt{6}\times\sqrt{8}=\sqrt{6\times{8}}=\sqrt{48}$

Simplify the product.

The product of $(12\sqrt{6})(\sqrt{8})$ is $12\sqrt{48}$

$12\sqrt{48}$ can be simplified further because $48$ has a perfect square factor.

$12\sqrt{48}=12\sqrt{16\times{3}}=12\sqrt{16}\times\sqrt{3}=12\times{4}\times\sqrt{3}=48\sqrt{3}$

The simplified product of $(12\sqrt{6})(\sqrt{8})=48\sqrt{3}$

Example 4: multiplying radicals using distributive property

Multiply the radical expression $\sqrt{2}(4\sqrt{6}-7)$ .

Multiply the coefficients together and the radicands together.

Use the distributive property to multiply.

First calculate $\sqrt{2}\times{4}\sqrt{6}.$

Multiply the coefficients together: $1\times{4}=4$

Multiply the radicals together: $\sqrt{2}\times\sqrt{6}=\sqrt{12}$

So, $\sqrt{2}\times{4}\sqrt{6}=4\sqrt{12}$

Next, multiply $\sqrt{2}\times(- \; 7)$

Multiply the whole numbers: $1\times(- \; 7)=- \; 7$

There are not two radicals to multiply together, so include the radical in the answer.

$\sqrt{2} \times(- \; 7)=- \; 7 \sqrt{2}$

$\sqrt{2}(4 \sqrt{6}-7)=4 \sqrt{12}-7 \sqrt{2}$

Simplify the product.

The product, $4\sqrt{12}-7\sqrt{2}$ can be simplified further because $\sqrt{12}$ can be broken down. $12$ has a perfect square factor.

\begin{aligned}4\sqrt{12}&=4\times\sqrt{4\times{3}} \\\\ &=4\times\sqrt{4}\times\sqrt{3} \\\\ &=4\times{2}\times\sqrt{3} \\\\ &=8\sqrt{3} \end{aligned}

The simplified product of $\sqrt{2}(4\sqrt{6}-7)=8\sqrt{3}-7\sqrt{2}$

Example 5: multiply using distributive property

Multiply the radical expression $(\sqrt{10}+3)(\sqrt{2}-6).$

Multiply the coefficients together and the radicands together.

Use the distributive property.

First multiply, $\sqrt{10}\times\sqrt{2}$ and $\sqrt{10}\times(- \; 6).$ Remember to multiply the coefficients to each other, the radicands to each other, and in this case the whole numbers to each other.

$\sqrt{10}\times\sqrt{2}=\sqrt{20}$

$\sqrt{10}\times(- \; 6)=- \; 6\sqrt{10}$

Next multiply, $3 \times\sqrt{2}$ and $3\times(- \; 6)$

$3\times\sqrt{2}=3\sqrt{2}$

$3\times(- \; 6)=- \; 18$

Therefore, $(\sqrt{10}+3)(\sqrt{2}-6)=\sqrt{20}-6\sqrt{10}+3\sqrt{2}-18$

Simplify the product.

The product, $\sqrt{20}-6\sqrt{10}+3\sqrt{2}-18$ can be simplified further because $\sqrt{20}$ can be broken down. $20$ has a perfect square factor.

\begin{aligned}\sqrt{20}&=\sqrt{4\times{5}} \\\\ &=\sqrt{4}\times\sqrt{5} \\\\ &=2\sqrt{5} \end{aligned}

The simplified product of $(\sqrt{10}+3)(\sqrt{2}-6)=2\sqrt{5}-6\sqrt{10}+3\sqrt{2}-18$

Example 6: squaring a radical expression

Simplify the radical expression $(\sqrt{11}+5)^2.$

Multiply the coefficients together and the radicands together.

$(\sqrt{11}+5)^2$ means to multiply that radical expression to itself.

So, $(\sqrt{11}+5)^2=(\sqrt{11}+5)(\sqrt{11}+5),$ which means you have to use the distributive property.

First multiply $\sqrt{11}\times\sqrt{11}$ and $\sqrt{11}\times{5}.$ Remember to multiply coefficients to each other, radicands to each other and whole numbers to each other.

$\sqrt{11}\times\sqrt{11}=\sqrt{121}=11$

$\sqrt{11} \times 5=5 \sqrt{11}$

Then multiply, $5\times\sqrt{11}$ and $5\times{5}.$

$5\times\sqrt{11}=5\sqrt{11}$

$5\times{5}=25$

(\sqrt{11}+5)^2=11+5\sqrt{11}+5\sqrt{11}+25

Simplify the product.

The product $11+5 \sqrt{11}+5 \sqrt{11}+25$ can be further simplified.

$5\sqrt{11}+5\sqrt{11}$ can be added together because they have like radicals. So, $5\sqrt{11}+5\sqrt{11}=10\sqrt{11}$

\begin{aligned}(\sqrt{11}+5)^2&=11+10\sqrt{11}+25 \\\\ &=10\sqrt{11}+36 \end{aligned}

Teaching tips for multiplying radicals

Remind students that there is always a $1$ in front of the radical symbol.

Instead of assigning worksheets to students, have them practice skills of simplifying radical expressions, multiplying radical expressions, and dividing radical expressions using digital game-play or having students do scavenger hunts around the classroom.

Use online tutorials for struggling learners where they can view full videos on step-by-step processes on how to simplify and add radicals.

Have students that struggle with math facts, use the square root function on a handheld or digital calculator.

Easy mistakes to make

Thinking that you can only multiply radical expressions if the radicals are the same
Radicals can be multiplied together, whether they are the same or not.

$\sqrt{3}\times\sqrt{3}=\sqrt{9}=3$

$\sqrt{5}\times\sqrt{2}=\sqrt{10}$

$\begin{matrix} \end{matrix}$

Forgetting to simplify each radical fully
For example, when multiplying, $\sqrt{6}\times\sqrt{3}=\sqrt{18},$ simplify $\sqrt{18}$ because it has a perfect square factor.

$\begin{aligned}\sqrt{18}&=\sqrt{9\times{2}} \\\\ &=\sqrt{9}\times\sqrt{2} \\\\ &=3\sqrt{2} \end{aligned}$

Practice multiplying radicals questions

30

15\sqrt{4}

8\sqrt{2}

15

Multiply the coefficients to each other and the radicands to each other.

5\times{3}=15

\sqrt{2}\times\sqrt{2}=\sqrt{4}=2

Simplify the product.

5 \sqrt{2} \times 3 \sqrt{2}=15 \times 2=30

30\sqrt{12}

60\sqrt{5}

30\sqrt{20}

26\sqrt{5}

Multiply the coefficients to each other and the radicands to each other.

10 \times 3=30

\sqrt{2} \times \sqrt{10}=\sqrt{20}

$\sqrt{20}$ can be simplified because it has a perfect square factor.

\begin {aligned} \sqrt{20}&=\sqrt{4 \times 5} \\\\ &=\sqrt{4} \times \sqrt{5} \\\\ &=2\sqrt{5} \end{aligned}

Simplify the product

30\times{2}\sqrt{5}=60\sqrt{5}

10\sqrt{2}\times{3}\sqrt{10}=60\sqrt{5}

2\sqrt{10}+\sqrt{3}

2\sqrt{21}+1

\sqrt{42}+\sqrt{3}

2\sqrt{21}+\sqrt{3}

Multiply using the distributive property.

Multiplying radicals 7 US

First multiply $\sqrt{3}\times{2}\sqrt{7}$ and then $\sqrt{3}\times{1}$

\sqrt{3}\times{2}\sqrt{7}=2\sqrt{21}

\sqrt{3}\times{1}=1\sqrt{3}=\sqrt{3}

\sqrt{3}(2\sqrt{7}+1)=2\sqrt{21}+\sqrt{3}

– \; 4\sqrt{10}-7\sqrt{5}

– \; 4\sqrt{7}-7\sqrt{5}

4\sqrt{10}+7

– \; 4\sqrt{10}+7\sqrt{5}

Use the distributive property to multiply the radical expression.

First multiply $– \; \sqrt{5}\times{4}\sqrt{2}$ and then $– \; \sqrt{5}\times{7}$

Multiply coefficients together and radicands together.

– \; \sqrt{5}\times{4}\sqrt{2}=- \; 4\sqrt{10}

– \; \sqrt{5}\times{7}=- \; 7\sqrt{5}

– \; \sqrt{5}(4\sqrt{2}+7)=- \; 4\sqrt{10}-7\sqrt{5}

4\sqrt{7}-24\sqrt{2}+3\sqrt{5}

4\sqrt{10}+24\sqrt{2}-3\sqrt{5}-18

– \; 4\sqrt{10}-24\sqrt{2}+3\sqrt{5}+18

4\sqrt{10}-24\sqrt{2}-3\sqrt{5}-18

Use the distributive property to multiply $(4 \sqrt{2}-3)(\sqrt{5}+6).$

$(4 \sqrt{2}-3)(\sqrt{5}+6)$ multiply $4\sqrt{2}\times\sqrt{5}$ and $4\sqrt{2}\times{6}$

Multiplying radicals 8 US

First, multiply coefficients to each other and radicands to each other.

4\sqrt{2}\times\sqrt{5}=4\sqrt{10}

4\sqrt{2}\times{6}=24\sqrt{2}

– \; 3 \times \sqrt{5}=- \; 3 \sqrt{5}

– \; 3 \times 6=- \; 18

(4\sqrt{2}-3)(\sqrt{5}+6)=4\sqrt{10}+24\sqrt{2}-3\sqrt{5}-18

18-2\sqrt{7}

88-18\sqrt{7}

81+2\sqrt{7}

81+18\sqrt{7}

\left(9-\sqrt{7}\right)^2=(9-\sqrt{7})(9-\sqrt{7})

Use the distributive property to multiply the expressions together.

Multiplying radicals 9 US

Multiply coefficients with each other and the radicals with each other.

9\times{9}=81

9\times(-\sqrt{7})=- \; 9\sqrt{7}

– \; \sqrt{7}\times{9}=- \; 9\sqrt{7}

– \; \sqrt{7}\times-\sqrt{7}=\sqrt{49}=7

\left(9-\sqrt{7}\right)^2=81-9\sqrt{7}-9\sqrt{7}+7

You can combine the radicals with ‘like radicals’ so

– \; 9\sqrt{7}-9\sqrt{7}=- \; 18\sqrt{7}

\left(9-\sqrt{7}\right)^2=81-18\sqrt{7}+7=88-18\sqrt{7}

Multiplying radicals FAQs

Can you multiply cube roots together?

Yes, you can multiply cube roots or any root.

What is the process when dividing radical expressions?

Radical expressions can be divided. You will learn the process on dividing radicals in algebra $1$ and algebra $2$ or you can go to the “dividing radicals” webpage.

Is the process for solving radical equations the same as for solving quadratic equations?

The process is different. To solve simple radical equations, you need to isolate the radical part to be on one side of the equation. Then apply the correct exponent to both sides of the equation to get rid of the radical sign.

For example, if the root is a cube root, then the exponent or power each side of the equation needs to be raised to is $3.$

Can a rational expression contain a radical expression?

Yes, rational expressions are like fractions because they have a numerator and denominator. There can be a radical expression in either the numerator, denominator, or both, but the number value will be irrational.

How do the graphs of polynomial, radical, and exponential functions compare to one another?

The graphs of these functions are quite different. Polynomial functions are smooth curves with no breaks because the domain is all real numbers.

Exponential functions are smooth curves too because the domain is all real numbers but will have a horizontal asymptote. Radical functions are not necessarily classified as being smooth curves because the domain is not all real numbers.

Are functions with rational exponents considered to be radical functions?

Functions or expressions with rational exponents are considered to be radical functions because you can represent a root symbol with a rational exponent.

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