Meaning before procedure
Like radicals have the same index and the same simplified radicand. Their coefficients combine just like coefficients of like variable terms: a√m + b√m = (a+b)√m.
- Like radicals
- Same index and same simplified radicand.
- Coefficient
- The factor multiplying a radical.
- Unlike radicals
- Terms that cannot combine by addition/subtraction.
- Combine
- Add or subtract coefficients while preserving the radical part.
Interactive visual model
Change coefficients and see that the shared radical unit stays fixed.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Confirm each radical is simplified.
- Group terms with identical radical parts.
- Add or subtract their coefficients.
- Keep unlike radical terms separate.
Watch the reasoning, not just the answer
Example 1: Simplify 3√5 + 7√5.
Example 2: Simplify 8∛2 − 3∛2.
Example 3: Simplify 4√3 + 2√7 − √3.
Predict, decide, check, and revise
Build a valid reasoning path
Choose the decisions in the order a careful solver would use them for this concept.
The reasoning path changes as each decision is selected.
Check the foundation
Simplify 3√2 + (5)√2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify 7√3 + (-2)√3.
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Apply the method with support
Simplify 4√7 + (6)√7.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify 9√11 + (-5)√11.
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Simplify 2√5 + (8)√5.
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Work without step prompts
Simplify -3√2 + (10)√2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Simplify 12√13 + (-7)√13.
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Simplify 5√3 + (5)√3.
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Simplify 1√7 + (-6)√7.
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Demonstrate durable understanding
Simplify 8√2 + (3)√2.
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Simplify -4√5 + (-5)√5.
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Simplify 15√6 + (-9)√6.
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Simplify 7√10 + (11)√10.
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Simplify 20√3 + (-13)√3.
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Explain, compare, and revise
Simplify 6√11 − 2√3 + 5√11 + 7√3 and explain the grouping.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Group like radical parts: (6+5)√11 + (−2+7)√3 = 11√11 + 5√3. The two final radicals are unlike, so the expression is simplified.
The learner record reveals exact answers and model reasoning only after an attempt.