Radical Expressions • Topic 3 of 4

Rationalizing Denominators

Rationalizing removes a radical from a denominator. For a single root, multiply numerator and denominator by that root: 6/√3 = 6√3/3 = 2√3. For a denominator like √a − b, multiply by its conjugate √a + b, using the difference of squares to clear the root: 1/(√5 − 2) × (√5 + 2)/(√5 + 2) = (√5 + 2)/(5 − 4) = √5 + 2. The result is an equivalent expression with a rational denominator. The SAT expects simplified, rationalized answers, and the conjugate trick is the standard tool for binomial radical denominators.

✅ Solved examples

1. Rationalize 6/√3.
Multiply by √3/√3: 6√3/3 = 2√3.
2. Rationalize 1/√2.
Multiply by √2/√2: √2/2.
3. Rationalize 10/√5.
Multiply by √5/√5: 10√5/5 = 2√5.
4. Rationalize 1/(√5 − 2).
Multiply by the conjugate √5 + 2: (√5 + 2)/(5 − 4) = √5 + 2.

✏️ Practice — try these, take hints as needed

1. Rationalize 4/√2.
Multiply by √2/√2.
4√2/2.
—
2√2.
2. Rationalize 9/√3.
Multiply by √3/√3.
9√3/3.
—
3√3.
3. Rationalize 14/√7.
Multiply by √7/√7.
14√7/7.
—
2√7.
4. Rationalize 5/√5.
Multiply by √5/√5.
5√5/5.
—
√5.
5. Rationalize 1/(√3 − 1).
Multiply by conjugate √3 + 1.
Denominator: 3 − 1 = 2.
—
(√3 + 1)/2.

📝 Topic test — 8 questions

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