## Multiplication of Radicals

How to multiply radical expressions.

In this lesson, we are only going to deal with square roots only which is a specific type of radical expression with an index of [latex]\color{red}2[/latex]. If you see a radical symbol without an index explicitly written, it is understood to have an index of [latex]\color{red}2[/latex].

Below are the basic rules in multiplying radical expressions.

## Basic Rule on How to Multiply Radical Expressions

A radicand is a term inside the square root. We multiply radicals by multiplying their radicands together while keeping their product under the same radical symbol. What happens then if the radical expressions have numbers that are located outside?

We just need to tweak the formula above. But the key idea is that the product of numbers located outside the radical symbols remains outside as well.

Let’s go over some examples to see how these two basic rules are applied.

## Examples of How to Multiply Radical Expressions

Example 1 : Simplify by multiplying.

Multiply the radicands while keeping the product inside the square root.

The product is a perfect square since 16 = 4 · 4 = 4 2 , which means that the square root of [latex]\color{blue}16[/latex] is just a whole number.

Example 2 : Simplify by multiplying.

It is okay to multiply the numbers as long as they are both found under the radical symbol. After the multiplication of the radicands, observe if it is possible to simplify further.

Example 3 : Simplify by multiplying.

Take the number outside the parenthesis and distribute it to the numbers inside. We are just applying the distributive property of multiplication.

Next, proceed with the regular multiplication of radicals. Be careful here though. You can only multiply numbers that are inside the radical symbols. In the same manner, you can only numbers that are outside of the radical symbols.

When multiplying a number inside and a number outside the radical symbol, simply place them side by side.

Example 4 : Simplify by multiplying.

Similar to Example 3, we are going to distribute the number outside the parenthesis to the numbers inside. But make sure to multiply the numbers only if their “locations” are the same. That is, multiply the numbers outside the radical symbols independent from the numbers inside the radical symbols.

From here, I just need to simplify the products.

Example 5 : Simplify by multiplying.

Example 6 : Simplify by multiplying two binomials with radical terms.

This problem requires us to multiply two binomials that contain radical terms. Apply the FOIL method to simplify.

- F : Multiply the first terms.
- O : Multiply the outer terms.
- I : Multiply the inner terms.
- L : Multiply the last terms.

After applying the distributive property using the FOIL method , I will simplify them as usual.

Example 7 : Simplify by multiplying two binomials with radical terms.

Just like in our previous example, let’s apply the FOIL method to simplify the product of two binomials.

From this point, simplify as usual. Notice that the middle two terms cancel each other out.

Example 8 : Simplify by multiplying two binomials with radical terms.

Let’s solve this step-by-step:

- Multiply together using the FOIL method.
- Simplify the product.
- Simplify the signs.
- Add similar radicals.
- Simplify the square root of [latex]25[/latex].
- Add the numbers without radical symbols

Example 9 : Simplify by multiplying two binomials with radical terms.

- Expand the product of binomials using FOIL .
- Get the square roots of perfect square numbers which are [latex]\color{red}36[/latex] and [latex]\color{red}9[/latex].
- Find a perfect square factor for 24 .
- Break it down as a product of square roots.
- Simplify the square root of [latex]4[/latex].
- Subtract the similar radicals, and subtract also the numbers without radical symbols.

Example 10 : Simplify by multiplying.

We are going to multiply these binomials using the “matrix method”. Write the terms of the first binomial ( in blue ) in the left-most column, and write the terms of the second binomial ( in red ) on the top row.

Multiply the numbers of the corresponding grids. See the animation below.

Next, simplify the product inside each grid.

Finally, add all the products in all four grids, and simplify to get the final answer.

Example 11 : Simplify by multiplying.

Place the terms of the first binomial in the left-most column, and the terms of the second binomial on the top row. Then multiply the corresponding square grids.

Finally, add the values in the four grids, and simplify as much as possible to get the final answer.

You may also be interested in these related math lessons or tutorials:

Solving Radical Equations Simplifying Radical Expressions Adding and Subtracting Radical Expressions Rationalizing the Denominator

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## Multiplying and Dividing Radical Expressions

Adding and Subtracting Radical Expressions |

## Multiplying radical expressions

We multiply binomial expressions involving radicals by using the FOIL (First, Outer, Inner, Last) method.

Example 1: Multiply each of the following

Example 2: Multiply each of the following

Exercise 1: Multiply each of the following

$$ \color{blue}{\left( \sqrt5 - \sqrt3 \right) \left( \sqrt3 - \sqrt5\right) = } $$ | $ -2 $ | |

$ 2 $ | ||

$ 4 $ | ||

$ -8 $ |

$$ \color{blue}{\left( 2 \sqrt5 - \sqrt2 \right)^2 = } $$ | $ 2\sqrt{10} - 2 $ | |

$ \sqrt{10} - 11 $ | ||

$ 22 - 2\sqrt{10} $ | ||

$ 18 $ |

## Dividing Radical Expressions

A common way of dividing the radical expression is to have the denominator that contain no radicals. Dividing radical is based on rationalizing the denominator . Rationalizing is the process of starting with a fraction containing a radical in its denominator and determining fraction with no radical in its denominator.

Techniques for rationalizing the denominator are shown below.

CASE 1: Rationalizing denominators with one square roots

When you have one root in the denominator you multiply top and bottom by it.

Example 3: Rationalize each denominator

Exercise 2: Rationalize each denominator

$$ \color{blue}{\frac{12}{\sqrt3}} $$ | $ 12\sqrt3 $ | |

$ 6\sqrt3 $ | ||

$ 4\sqrt3 $ | ||

$ \sqrt3 $ |

$$ \color{blue}{\frac{24}{\sqrt{12}}} $$ | $ \sqrt{12} $ | |

$ 4\sqrt{3} $ | ||

$ 2\sqrt{24} $ | ||

$ \sqrt{2} $ |

CASE 2: Rationalizing Denominators with Cube Roots

Here you need to multiply the numerator and denominator by a number that will result in a perfect cube in the radicand in the denominator.

Example 4: Rationalize each denominator

Exercise 3: Rationalize each denominator

$$ \color{blue}{\frac{8}{\sqrt[\large{3}]{2}}} $$ | $ 4 \sqrt[\large{3}]{4} $ | |

$ 4 \sqrt[\large{3}]{2} $ | ||

$ 2 \sqrt[\large{3}]{4} $ | ||

$ 2 \sqrt[\large{3}]{2} $ |

$$ \color{blue}{\frac{8}{\sqrt[\large{3}]{4}}} $$ | $ 4 \sqrt[\large{3}]{2} $ | |

$ 4 \sqrt[\large{3}]{4} $ | ||

$ 2 \sqrt[\large{3}]{4} $ | ||

$ 2 \sqrt[\large{3}]{2} $ |

CASE 3: Rationalize denominators with binomials

In this case, you will need to multiply the denominator and numerator by the same expression as the denominator but with the opposite sign in the middle . This expression is called the conjugate of the denominator .

Example 5: Rationalize denominator in $ \frac{4}{\sqrt{5} + \sqrt{3}} $ .

In this example denominator is $ \color{blue}{\sqrt{5}} \color{red}{+} \color{blue}{\sqrt{3}} $ so we will multiply the denominator and numerator with $ \color{blue}{\sqrt{5}} \color{red}{-} \color{blue}{\sqrt{3}} $. Here is a complete solution:

Example 6: Rationalize denominator in $ \frac{\sqrt 8 - \sqrt 3}{\sqrt 6 - 2} $ .

In this example denominator is $ \sqrt 6 \color{red}{-} 2 $ so we will multiply the denominator and numerator by $ \sqrt 6 \color{red}{+} 2 $. The solution is:

Exercise 4: Rationalize each denominator

$$ \color{blue}{\frac{2}{\sqrt5 - \sqrt3}} $$ | $ \sqrt5 - \sqrt3 $ | |

$ \sqrt5 + \sqrt3 $ | ||

$ - \sqrt5 + \sqrt3 $ | ||

$ - \sqrt5 - \sqrt3 $ |

$$ \color{blue}{\frac{\sqrt{18}-\sqrt2 }{\sqrt7 - 2}} $$ | $ 2\left(\sqrt7 - \sqrt5 \right) $ | |

$ 2\left(\sqrt7 + \sqrt5 \right) $ | ||

$ 2\sqrt2\left(\sqrt7 - \sqrt5 \right) $ | ||

$ \sqrt2\left(\sqrt7 - \sqrt5 \right) $ |

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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).

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

This strategy can be applied to radical expressions.

## [FREE] Algebra Worksheet (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of algebra. 10+ questions with answers covering a range of 6th and 8th grade algebra topics to identify areas of strength and support!

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.

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.

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:

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,

## 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.

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

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

2 Simplify 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.

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

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

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: 12\times{1}=12

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

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) .

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}

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.

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).

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

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.

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.

(\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

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}

## 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} 3 × 3 = 9 = 3 5 × 2 = 10 ” role=”presentation” style=”font-size: 113%; text-align: center; position: relative;”>
- 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}

## Related Radicals lessons

- Subtracting radicals
- Rationalize the denominator
- Dividing radicals

## Practice multiplying radicals questions

1. Multiply the radicals, and be sure to simplify the product completely.

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

2. Multiply the radical expression, 10\sqrt{2}\times{3}\sqrt{10} .

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

Simplify the product

3. Multiply the radical expression and simplify the product.

Multiply using the distributive property.

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

4. Simplify the radical expression – \; \sqrt{5}(4\sqrt{2}+7).

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.

5. Simplify fully the radical expression (4 \sqrt{2}-3)(\sqrt{5}+6).

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}

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

6. Square the radical expression (9-\sqrt{7})^2.

Use the distributive property to multiply the expressions together.

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

You can combine the radicals with ‘like radicals’ so

## Multiplying radicals FAQs

Yes, you can multiply cube roots or any root.

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.

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.

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.

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.

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

## The next lessons are

- Radical functions
- Quadratic equation

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## [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!

## Privacy Overview

## Multiplying Radicals Worksheet

Students will practice multiplying square roots (ie radicals). This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges.

## Example Questions

Directions: Mulitply the radicals below. (Express your answer in simplest radical form)

## Challenge Problems

## Other Details

This is a 4 part worksheet:

- Part I Model Problems
- Part II Practice
- Part III Challenge Problems
- Part IV Answer Key
- Radical Reducer Rewrites any radical in simpliest radical form

## Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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Learn how to multiply radicals with this flashcard set that covers the basic rules and steps for simplifying the product of two radical expressions. You can also practice your skills with different examples and check your answers with the solutions provided.

Multiplying Cube Roots and Square Roots Learn with flashcards, games, and more — for free.

Study with Quizlet and memorize flashcards containing terms like How do you multiply radicals?, 3Sqrt(12)*Sqrt(6), Sqrt(5)*Sqrt(10) and more.

We can't simplify 3 4 and we can't combine the terms, because they don't have radicals with matching indices or radicands. This is the final simplified answer. How to Apply the FOIL Method to Multiply Radical Expressions. Multiphy: (4. x − 2 5)(2. x + 3 5) I have two __________ being multiplied by each other.

The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. Apply the distributive property, simplify each radical, and then combine like terms.

Slide. 2. Review: Common Problem Types. • To multiply radicals with like , use the product property of roots. • The commutative property, distributive property, and FOIL method can be used to multiply expressions containing radicals. • To multiply radicals with indices, write the radicals using rational exponents. indices.

Learn how to multiply radical expressions with an index. Get an understanding of the basics of radical expression multiplication!

Like radicals are radical expressions with the same index and the same radicand. We add and subtract like radicals in the same way we add and subtract like terms. We know that 3x + 8x is 11x. Similarly we add 3√x + 8√x and the result is 11√x. Let's think about adding like terms with variables as we do the next few examples.

Math lesson on multiplying and dividing radical expressions with examples, solutions and exercises.

Free multiplying radicals math topic guide, including step-by-step examples, free practice questions, teaching tips and more!

Students also viewed Dividing radicals assignments 12 terms chantal1501 Preview Multiplying Radicals Teacher 14 terms NWcoachFurfaro Preview Normal measurements 24 terms cwyatt30 Preview Chem 127 Terms 6 terms allen_emma2005 Preview NIHSS - Group A Teacher 6 terms SLP-USA Preview Chapter 11: Electric Current and Electric Circuits 14 terms ally ...

Slide. 2. Review: Common Problem Types. • To multiply radicals with like , use the product property of roots. • The commutative property, distributive property, and FOIL method can be used to multiply expressions containing radicals. • To multiply radicals with indices, write the radicals using rational exponents.

Objective Students will practice multiplying square roots (ie radicals). This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges.

Use this page to find the questions and answers for an assessment. This page is not to view a student's answers.

You learned about rewriting expressions involving radicals. Simplifying Radicals When simplifying, one of the factors must be a root. Adding and Subtracting Radicals Multiplying Radicals First, ensure radicands. Then, perform the First, the

Study with Quizlet and memorize flashcards containing terms like 24√5, -16x√11, 6√6 and more.

Multiplying Radicals question for 8th grad students. Find other quizzes for Mathematics and more on Quizizz for free!

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Express respectively explanation in simplest terms. All Subjects. Linear Sentences in Neat Variable. Linear Equations; Quizfrage: Linear Equations

You learned about rewriting expressions involving radicals. Simplifying Radicals When simplifying, one of the factors must be a root. Adding and Subtracting Radicals First, ensure radicands. Then, perform the . Multiplying Radicals First, the coefficients and the radicands. Then, simplify the expression if possible. perfect like operation

Slide 2 Lesson Question How is adding and subtracting radical expressions similar to combining like terms? Answer Review: Key Concepts • Like radicals have the

View Lecture Slides - Week003-Assignment-Kyle-Julian-C.-Libatique (2) - Copy.docx from TAX LAWS BAMM6202 at AMA University Online Education. MATH6100 - Calculus 1 (WEEK 3: GRAPH OF

Lesson Question How can you write a ratio of radical expressions in simplest form? Answer (Sample answer) You can simplify a ratio of radical expressions by applying the quotient property of radicals and rationalizing the denominator. Review: Key Concepts