Meaning before procedure
Radicals that look unlike may become like after perfect-power factors are extracted. The correct order is simplify each radical, then group matching radical parts, then combine coefficients.
- Hidden like radicals
- Different original radicands that simplify to the same radical part.
- Term-by-term simplification
- Simplify each radical without crossing addition signs.
- Coefficient product
- Multiply original and extracted coefficients.
- Final collection
- Combine only after forms are simplified.
Interactive visual model
Simplify hidden like radicals, then combine their coefficients.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Simplify each radical term independently.
- Multiply extracted factors by existing coefficients.
- Identify matching simplified radical parts.
- Combine coefficients and verify no further simplification remains.
Watch the reasoning, not just the answer
Example 1: Simplify √8 + √18.
Example 2: Simplify 3√12 − √27.
Example 3: Simplify 2√50 + √8 − 3√2.
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 √8+√18.
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Simplify √12+√27.
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Apply the method with support
Simplify √20+√45.
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Simplify 2√8+√50.
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Simplify 3√12−√27.
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Work without step prompts
Simplify √75−√12.
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Simplify 2√18+3√8.
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Simplify √48+√75.
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Simplify 3√20−2√45.
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Demonstrate durable understanding
Simplify √98−√8.
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Simplify 2√27+√12.
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Simplify √32+2√50−3√2.
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Simplify 4√12−√75.
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Simplify 2√45+√80.
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Explain, compare, and revise
A rectangle has side lengths √75 and 2√12. Find its perimeter in simplest radical form.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Perimeter P = 2(√75 + 2√12). Simplify: √75 = 5√3 and 2√12 = 4√3. Then P = 2(9√3) = 18√3 units.
The learner record reveals exact answers and model reasoning only after an attempt.