These worksheets will teach your students how to balance various algebraic equations that contain radicals.

Radicals often complicate otherwise simple problems. Especially when there are some form operations involved. These worksheets will look at problems that are simple and some that are difficult and will make you look at it several times and maybe even require you take a break before attempting the next problem. The work here starts by learning simplify radical values and progresses to processing operations. We will start with set workable problems. As we progress, we will move to more complex problems that require several steps to complete them. If you remind yourself of the general rules, these will go down easy. The two lessons in this section explain how to handle equations including one or more radical numbers.

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This lesson will walk you through an entire problem and then give the opportunity to practice on 2 problems. You will simplify this radical equation by following the steps below: 4√16

This worksheet will include problems that cover how all four operators are used in conjuction with this skill. For each problem, perform the indicated operations and simplify the answers. Example: 5√9 + 12√9

Worksheet 2

Perform the indicated operations using the rules we have learned so far. Example: 2√16 / 8√16

Quiz

For each problem, perform the indicated operations and then simplify the answers. Afterward, be sure check your answers and record your total score below. Example: 20√9 + 10√9

We progress toward slightly more complex operations with more pieces to put together. Example: 2√2 + 4√2 - 3√2

Intermediate Worksheet 1

Simplify each of the following radical expressions. In most problems there are at least 2 operators that you must navigate. Example: 2√9 + 6√16 - 9√25

Intermediate Worksheet 2

Don't let the brackets get in the way. The operations still follow the same procedure we have been using. Example: (6 + √5)∙(2- √5)

Review Sheet

This is more of reduction activity than anything else. Simplify the following: 6√4 + 4√4 - 3√4

Time to see what you have learned from this section. Put those lessons to work for you if you are having trouble. Example: √16∙4√(25)

Working with Parentheses Do Now

It is a great idea to do this as a class after having introduced the topic. These three problems overly use parentheses. Example: (6 - √25)2

Radical equations are not only essential in high school mathematics but also in many advanced studies. It is essential to learn how to process types of problems because they have a large footprint in more advanced applications of math. We will explain what radicals are and how to process operations with them.

Any equation that has a radical in it is called a radical equation. Radicals are math exponents that are symbolized by '√.'

Have you seen a number on the left of the radical in a small size? That number is called a degree or index number. The number being rooted is called the radicand.

The index number states how many times a number needs to be multiplied to equal the radicand. For example, √36 would be 6 x 6, and ∛216 would be 6 x 6 x 6.

An equation written below the vinculum is called a radical equation. Likewise, an expression written below the vinculum is called a radical expression.

You can quickly solve equations involving radicals if you know the rules. Here are some of the core concepts behind these rules:

The radical is a square root (√) if there is no index number on the left of the radical.
If the number being rooted is positive, the result should also be positive. The result should be negative if the number being rooted is negative.
If the radicand is negative and an index number is an even number, the resultant should be an irrational number.
If the index number of a radical for two radicands being multiplied is the same, the radicands will multiply while the radical will remain the same. For example, ∛16 x ∛5 = ∛80.
If the index number of a radical for two radicands being divided is the same, the radicands will divide while the radical remains the same. For example, ∛16 / ∛4 = ∛4
The radicand can be split under two similar radicals. For example, √20 = √5 x √4
A radical can be removed if it is powered to the same number as the index number. For example (∛6)3 = 6

How to Process These Types of Operations

To add or subtract radical equations, you must separate like term variables. The thing to focus on is the radicand and the index. If they are the same, you can add or subtract the two. Let's look at an example:

Example 1: 5√6 + 8√6

Since the index number for both is a square root and the radicands are both six, we can add the exponents,

5√6 + 8√6 = √6(5 + 8) = 13√6

Example 2: 8√6 – 5√6

Since the index number for both is a square root and the radicands are both six, we can add the exponents,

8√6 – 5√6 = √6(8 – 5) = 3√6

Multiplication and Division

To multiply, you need separate the term variables. The thing to focus on is the index. If the index is the same, you can multiply the two. Let's look at an example.

Example 1: √8 x √5

Since the index is the same, we can multiply the radicands.

√8 x √5 = √(8 x 5) = √40

To divide radical equations, you need to rewrite the radicand as a product of different factors. Let's look at an example,

Example 2: √48/9

√48/9 = √48 / √9 = √(4 x 4 x 3 / √(3 x 3) = √(42 x 3) / √32 = √42 x √3 / √32 = (4 x √3) / 3