Teacher Guide: Adding and Subtracting Radicals

Learn how to combine like radicals using addition and subtraction, just like combining like terms.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Radicals. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Identify like radicals based on index and radicand
Add and subtract like radicals by combining coefficients
Simplify radicals before attempting to combine them
Recognize when radicals cannot be combined
Apply radical addition and subtraction to geometric problems

Prerequisites

• Understanding of square roots and perfect squares
• Ability to simplify radicals
• Familiarity with combining like terms in algebra
• Basic understanding of the commutative property

Discussion Starters

1. Why do you think we can only add like radicals? What would go wrong if we added the radicands?
2. How is adding radicals similar to adding like terms with variables?
3. If , can you find other pairs that also equal ?
4. Why is it important to simplify radicals before trying to add them?

Common Misconceptions

Thinking

Not recognizing hidden like radicals

Changing the radicand when combining

Differentiation Ideas

For Struggling Students:

• Use color-coding to highlight like radicals
• Start with radicals that are already simplified
• Provide a reference chart of perfect squares
• Use the analogy to like terms extensively ( is like )

For On-Level Students:

• Include problems requiring simplification first
• Mix problems with both like and unlike radicals
• Introduce geometric application problems
• Have students create their own addition problems

For Advanced Students:

• Include cube roots and higher index radicals
• Work with variable radicands
• Combine with multiplication of radicals
• Solve equations involving radical addition

Standards Alignment

HSN.RN.A.2 (CCSS.MATH.CONTENT.HSN.RN.A.2)
Rewrite expressions involving radicals and rational exponents using the properties of exponents
A.SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)
Use the structure of an expression to identify ways to rewrite it

Lesson Resources

visualLike Radicals Sorting Activity
Students sort radical expressions into groups of like radicals
activitySimplify Then Combine
Practice identifying when simplification reveals like radicals
worksheetPerimeter Problems with Radicals
Real-world applications involving adding radical side lengths

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

To add or subtract radicals, they must be like radicals (same index and same radicand). Like radicals work just like like terms in algebra.

Like radicals have:

The same index (both square roots, both cube roots, etc.)
The same radicand (the number under the radical)

Think of it like combining apples: 3 apples + 7 apples = 10 apples.

Key Rule: You can only add or subtract the coefficients (numbers in front). The radical part stays the same.

Worked Examples

Simplify:

Check if radicals are like

Both have - same index (2) and same radicand (3) → Like radicals

Add the coefficients

→ 13

Keep the radical part

→

Answer:

Common Mistakes

Adding radicands:

Why it's wrong: You cannot add numbers under different radicals. This is like saying 1 apple + 1 orange = 2 apporanges!

Correct: cannot be simplified. They are unlike radicals.

Forgetting to simplify first:

Why it's wrong: If you simplify first, you can combine:

Correct: Always simplify radicals first, then check if they become like radicals.

Combining coefficients AND radicands:

Why it's wrong: Only the coefficients change. The radicand stays the same.

Correct:

Ignoring the coefficient of 1:

Why it's wrong: means . So , not just .

Correct:

Why It Matters

Adding and subtracting radicals appears throughout mathematics and science:

Geometry: Calculating perimeters when sides involve square roots, like meters
Physics: Combining measurements involving roots in wave calculations
Engineering: Simplifying expressions in structural calculations
Finance: Some risk calculations involve combining radical expressions

Mastering this skill is essential for solving equations with radicals, working with the Pythagorean theorem, and advancing in algebra and calculus.

Real World Applications

Perimeter of a Triangle

When finding perimeters of shapes with sides expressed as radicals, you need to add them.

Example:

A triangle has sides m, m, and m. The perimeter is meters.

1Try It Yourself

A garden has a triangular shape with sides of meters, meters, and meters.

What is the perimeter of the garden in simplest form?

Step 1: Write the mathematical expression

First simplify each side, then add:

Construction and Architecture

Architects often work with diagonal measurements that involve square roots.

Example:

Two roof sections have lengths meters and meters. Combined length: meters.

2Try It Yourself

A building has two diagonal supports: one is meters and the other is meters.

What is the total length of diagonal supports needed?

Step 1: Write the mathematical expression

Simplify each radical first:

Key Takeaways

1Like radicals have the same index and the same radicand (e.g., and )
2Add or subtract only the coefficients, keep the radical part unchanged
3Always simplify radicals first to find hidden like radicals (e.g., )
4Unlike radicals cannot be combined (e.g., stays as is)
5Think of radicals like variables: is like

Frequently Asked Questions

Can I add and ?

Yes, but simplify first! and , so . Note this is NOT !

How do I know if radicals are like radicals?

Check two things: (1) same index (square root, cube root, etc.) and (2) same radicand (number under the radical). and are like. and are not.

What if there's no coefficient written?

No coefficient means the coefficient is 1. So . When adding , you get .

Can I ever combine unlike radicals?

Only if they simplify to like radicals. For example, looks unlike, but simplifies to .

Glossary

Radical: An expression containing a root symbol, such as (square root) or (cube root)
Radicand: The number or expression under the radical sign. In , the radicand is 5
Index: The small number indicating the type of root. Square roots have index 2 (often not written), cube roots have index 3
Like radicals: Radicals with the same index and the same radicand. and are like radicals
Coefficient: The number multiplied by the radical. In , the coefficient is 4
Simplify a radical: Rewrite with the smallest possible radicand, removing perfect square factors