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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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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 12+27=53\sqrt{12} + \sqrt{27} = 5\sqrt{3}, can you find other pairs that also equal 535\sqrt{3}?
  • 4. Why is it important to simplify radicals before trying to add them?
Common Misconceptions

Thinking a+b=a+b\sqrt{a} + \sqrt{b} = \sqrt{a+b}

Remediation: Use numerical examples: 9+16=3+4=7\sqrt{9} + \sqrt{16} = 3 + 4 = 7, but 25=5\sqrt{25} = 5. They are not equal!

Not recognizing hidden like radicals

Remediation: Emphasize the mantra: SIMPLIFY FIRST. Show 8+2\sqrt{8} + \sqrt{2} becomes 22+2=322\sqrt{2} + \sqrt{2} = 3\sqrt{2}

Changing the radicand when combining

Remediation: Use the analogy: 3 apples + 2 apples = 5 apples (not 5 oranges). The type stays the same, only the count changes.

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 (5\sqrt{5} is like xx)

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)
35+75=1053\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}
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.
ax+bx=(a+b)xa\sqrt{x} + b\sqrt{x} = (a + b)\sqrt{x}
ax−bx=(a−b)xa\sqrt{x} - b\sqrt{x} = (a - b)\sqrt{x}

Worked Examples

Simplify: 43+934\sqrt{3} + 9\sqrt{3}
1

Check if radicals are like

Both have 3\sqrt{3} - same index (2) and same radicand (3) → Like radicals
2

Add the coefficients

4+9=134 + 9 = 13
3

Keep the radical part

13⋅313 \cdot \sqrt{3} → 13313\sqrt{3}

Common Mistakes

Adding radicands: 3+5=8\sqrt{3} + \sqrt{5} = \sqrt{8}

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

Correct: 3+5\sqrt{3} + \sqrt{5} cannot be simplified. They are unlike radicals.

Forgetting to simplify first: 8+2=8+2\sqrt{8} + \sqrt{2} = \sqrt{8} + \sqrt{2}

Why it's wrong: If you simplify 8=22\sqrt{8} = 2\sqrt{2} first, you can combine: 22+2=322\sqrt{2} + \sqrt{2} = 3\sqrt{2}

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

Combining coefficients AND radicands: 35+25=5103\sqrt{5} + 2\sqrt{5} = 5\sqrt{10}

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

Correct: 35+25=553\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}

Ignoring the coefficient of 1: 7+27=27\sqrt{7} + 2\sqrt{7} = 2\sqrt{7}

Why it's wrong: 7\sqrt{7} means 171\sqrt{7}. So 1+2=31 + 2 = 3, not just 22.

Correct: 7+27=17+27=37\sqrt{7} + 2\sqrt{7} = 1\sqrt{7} + 2\sqrt{7} = 3\sqrt{7}

Why It Matters

Adding and subtracting radicals appears throughout mathematics and science:
  • Geometry: Calculating perimeters when sides involve square roots, like 32+523\sqrt{2} + 5\sqrt{2} 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 12\sqrt{12} m, 27\sqrt{27} m, and 48\sqrt{48} m. The perimeter is 23+33+43=932\sqrt{3} + 3\sqrt{3} + 4\sqrt{3} = 9\sqrt{3} meters.

1Try It Yourself

A garden has a triangular shape with sides of 18\sqrt{18} meters, 50\sqrt{50} meters, and 8\sqrt{8} 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 535\sqrt{3} meters and 232\sqrt{3} meters. Combined length: 737\sqrt{3} meters.

2Try It Yourself

A building has two diagonal supports: one is 75\sqrt{75} meters and the other is 27\sqrt{27} 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., 353\sqrt{5} and 757\sqrt{5})
  • 2Add or subtract only the coefficients, keep the radical part unchanged
  • 3Always simplify radicals first to find hidden like radicals (e.g., 8=22\sqrt{8} = 2\sqrt{2})
  • 4Unlike radicals cannot be combined (e.g., 3+5\sqrt{3} + \sqrt{5} stays as is)
  • 5Think of radicals like variables: 35+25=553\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} is like 3x+2x=5x3x + 2x = 5x

Frequently Asked Questions

Can I add 4\sqrt{4} and 9\sqrt{9}?

Yes, but simplify first! 4=2\sqrt{4} = 2 and 9=3\sqrt{9} = 3, so 4+9=2+3=5\sqrt{4} + \sqrt{9} = 2 + 3 = 5. Note this is NOT 13\sqrt{13}!

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). 7\sqrt{7} and 7\sqrt{7} are like. 7\sqrt{7} and 11\sqrt{11} are not.

What if there's no coefficient written?

No coefficient means the coefficient is 1. So 5=15\sqrt{5} = 1\sqrt{5}. When adding 5+35\sqrt{5} + 3\sqrt{5}, you get 15+35=451\sqrt{5} + 3\sqrt{5} = 4\sqrt{5}.

Can I ever combine unlike radicals?

Only if they simplify to like radicals. For example, 12+27\sqrt{12} + \sqrt{27} looks unlike, but simplifies to 23+33=532\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}.

Glossary

Radical
An expression containing a root symbol, such as x\sqrt{x} (square root) or x3\sqrt[3]{x} (cube root)
Radicand
The number or expression under the radical sign. In 5\sqrt{5}, 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. 373\sqrt{7} and 575\sqrt{7} are like radicals
Coefficient
The number multiplied by the radical. In 434\sqrt{3}, the coefficient is 4
Simplify a radical
Rewrite with the smallest possible radicand, removing perfect square factors

More in This Topic

Introduction to Radicals

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Simplifying Square Roots

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Multiplying Radicals

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Dividing Radicals

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