Meaning before procedure
A perfect nth power is produced by raising an integer to n. Odd roots accept positive or negative radicands. In the real-number system, an even root requires a nonnegative radicand.
- Perfect square
- An integer of the form k².
- Perfect cube
- An integer of the form k³.
- Even index
- An index such as 2 or 4; a real radicand must be nonnegative.
- Odd index
- An index such as 3 or 5; negative radicands are allowed.
Interactive visual model
Compare even and odd powers to see why index parity controls real roots.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Read the index.
- Check the radicand's sign.
- Match the radicand to a perfect power.
- Raise the proposed root to the index to verify.
Watch the reasoning, not just the answer
Example 1: Evaluate √144.
Example 2: Evaluate ∛(−125).
Example 3: Classify ⁴√(−16) over the reals.
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
Which number is a perfect square?
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Which number is a perfect cube?
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Apply the method with support
Evaluate √121.
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Evaluate ∛(−216).
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Which has no real value?
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Work without step prompts
Why can ∛(−8) be real?
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Evaluate ⁴√16.
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Which expression equals −3?
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What restriction applies to an even real root?
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Demonstrate durable understanding
Which index permits a negative radicand over the reals?
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Evaluate ⁵√32.
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Evaluate √0.
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Which check verifies ⁴√81 = 3?
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What does index parity determine first?
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Explain, compare, and revise
Compare ∛(−32) and ⁴√(−32) in the real-number system.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
The cube root has an odd index, so ∛(−32) is real, although it is not an integer. The fourth root has an even index and a negative radicand, so it is not a real number.
The learner record reveals exact answers and model reasoning only after an attempt.