Meaning before procedure
The conjugate of a + b is a − b. Their product is a² − b², so middle radical terms cancel. This difference-of-squares structure rationalizes denominators such as 3 + √2.
- Conjugates
- Binomials with the same terms and opposite middle signs.
- Difference of squares
- (a+b)(a−b) = a²−b².
- Cancellation
- Opposite middle terms sum to zero.
- Equivalent fraction
- Created by multiplying by conjugate/conjugate.
Interactive visual model
Switch the sign to form a conjugate pair and observe the difference-of-squares denominator.
Change one control at a time and describe what changes—and what stays invariant.
A dependable four-step method
- Identify the denominator's conjugate.
- Multiply numerator and denominator by it.
- Use difference of squares in the denominator.
- Distribute/simplify the numerator and reduce if possible.
Watch the reasoning, not just the answer
Example 1: Rationalize 1/(3 + √2).
Example 2: Rationalize 2/(√5 − 1).
Example 3: Rationalize 1/(2√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
Rationalize 1/(2+√3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/(3+√2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Apply the method with support
Rationalize 1/(3−√2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 2/(√5−1).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/(√3+1).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Work without step prompts
Rationalize 1/(√5+2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 3/(4−√7).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 2/(5+√3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/(2√2+1).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Demonstrate durable understanding
Rationalize 1/(3+2√2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 4/(√7−√3).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 2/(√6+√2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 1/(4+√5).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Rationalize 5/(3−√2).
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Explain, compare, and revise
Rationalize 3/(4 − √7) and show where cancellation occurs.
Guided self-check
Judge the locked first attempt against every criterion.
Model reasoning
Multiply by (4 + √7)/(4 + √7). The denominator becomes 4² − (√7)² = 16 − 7 = 9. The numerator is 12 + 3√7, so the fraction simplifies to (4 + √7)/3.
The learner record reveals exact answers and model reasoning only after an attempt.