Meaning before procedure
Rationalizing multiplies numerator and denominator by a factor that makes the denominator a perfect power. For 1/√a, multiply by √a/√a to obtain √a/a.
- Rational denominator
- A denominator containing no radical.
- Form of 1
- An equal numerator and denominator that preserves value.
- Completing a power
- Supplying missing factors needed for a perfect nth power.
- Restriction
- The original denominator cannot equal zero.
Interactive visual model
Build the missing factor that turns a radical denominator into a perfect power.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Inspect the denominator and index.
- Choose the factor that completes a perfect power.
- Multiply numerator and denominator.
- Simplify and preserve original restrictions.
Watch the reasoning, not just the answer
Example 1: Rationalize 1/√5.
Example 2: Rationalize 3/(2√7).
Example 3: Rationalize 1/∛4.
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
Rationalize 1/√2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/√3.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Apply the method with support
Rationalize 1/√5.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 2/√7.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 3/√2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Work without step prompts
Rationalize 5/(2√3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 7/(3√5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 4/√11.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/∛2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Demonstrate durable understanding
Rationalize 1/∛4.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 2/∛3.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 3/∛9.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/⁴√2.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/⁴√8.
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Explain, compare, and revise
Rationalize 5/∛(9x), assuming x ≠ 0 and the real expression is defined.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
The radicand 9x = 3²x needs a factor 3x² to make 27x³. Multiply by ∛(3x²)/∛(3x²). The result is 5∛(3x²)/(3x).
The learner record reveals exact answers and model reasoning only after an attempt.