Surds & Indices • Topic 3 of 4

Rationalization

Rationalization clears surds from a denominator so an expression becomes computable. For a single surd, multiply top and bottom by that surd: 1/√5 = √5/5. For a two-term denominator like √a + √b, multiply by its conjugate √a − √b, because (√a + √b)(√a − √b) = a − b removes both roots. So 1/(√7 − √3) = (√7 + √3)/[(√7)² − (√3)²] = (√7 + √3)/4. This is the engine behind a classic CAT pattern: a long sum 1/(√1+√2) + 1/(√2+√3) + … telescopes after rationalizing, since each term becomes √(n+1) − √n and the middle terms cancel, leaving just the last minus the first. Always simplify the resulting numerator and remember the denominator a − b can be negative — keep track of the sign so you do not flip the whole fraction by accident.

✅ Solved examples

1. Rationalize 1/(√7 − √3).
Multiply by (√7 + √3): top = √7 + √3, bottom = 7 − 3 = 4. Result = (√7 + √3)/4.
2. Rationalize 6/√3.
Multiply top and bottom by √3: 6√3/3 = 2√3.
3. Simplify (√5 + √2)/(√5 − √2).
Multiply by (√5 + √2): top = (√5 + √2)² = 5 + 2 + 2√10 = 7 + 2√10; bottom = 5 − 2 = 3. Result = (7 + 2√10)/3.
4. Evaluate 1/(√2 + 1) + 1/(√3 + √2).
Each rationalizes: 1/(√2+1) = √2 − 1; 1/(√3+√2) = √3 − √2. Sum = √3 − 1.

✏️ Practice — try these, take hints as needed

1. Rationalize 1/(√5 + √2).
Multiply by the conjugate √5 − √2.
Bottom = 5 − 2.
Divide by 3.
(√5 − √2)/3
2. Rationalize 10/√5.
Multiply top and bottom by √5.
10√5 / 5.
Simplify.
2√5
3. Simplify 1/(3 − √7).
Conjugate is 3 + √7.
Bottom = 9 − 7 = 2.
Write the result.
(3 + √7)/2
4. Evaluate 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4).
Rationalize each term.
Each becomes √(n+1) − √n.
Telescopes to √4 − √1.
1
5. Simplify (√3 − 1)/(√3 + 1).
Multiply by (√3 − 1).
Top = (√3 − 1)² = 4 − 2√3.
Bottom = 3 − 1 = 2.
2 − √3

📝 Topic test — 8 questions

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