Meaning before procedure
To simplify √n, factor n as a perfect square times a leftover factor. Use √(ab) = √a·√b for nonnegative factors, then evaluate the perfect-square root.
- Perfect-square factor
- The largest square dividing the radicand is usually most efficient.
- Simplified radical
- No perfect-square factor greater than 1 remains inside.
- Coefficient
- The number extracted in front of the radical.
- Product property
- √(ab) = √a√b for a,b ≥ 0.
Interactive visual model
Choose a perfect-power factor and watch it move outside the radical.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Find the largest perfect-square factor.
- Rewrite as a product under the radical.
- Take the root of the square factor.
- Confirm the remaining radicand has no square factor.
Watch the reasoning, not just the answer
Example 1: Simplify √72.
Example 2: Simplify √180.
Example 3: Simplify 3√48.
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.
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Simplify √12.
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Apply the method with support
Simplify √18.
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Simplify √20.
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Simplify √27.
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Work without step prompts
Simplify √32.
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Simplify √45.
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Simplify √48.
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Simplify √50.
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Demonstrate durable understanding
Simplify √72.
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Simplify √75.
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Simplify √98.
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Simplify √108.
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Simplify √180.
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Explain, compare, and revise
Simplify 2√300 and explain why the result is fully simplified.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
√300 = √(100·3) = 10√3, so 2√300 = 20√3. The remaining radicand 3 has no perfect-square factor greater than 1.
The learner record reveals exact answers and model reasoning only after an attempt.