Radical Expressions • Topic 1 of 4

Simplifying Radicals

To simplify a square root, factor out the largest perfect square: √72 = √(36 · 2) = 6√2. The product rule √(ab) = √a·√b lets you split off the perfect-square part. A radical is fully simplified when the number under the root has no perfect-square factor other than 1. For fractions, simplify the numerator and denominator roots separately, √(9/16) = 3/4. With variables, √(x²) = x for x ≥ 0. This skill underlies adding radicals (only like radicals combine) and is a frequent SAT step, so memorising small perfect squares speeds it up.

✅ Solved examples

1. Simplify √50.
√50 = √(25 · 2) = 5√2.
2. Simplify √48.
√48 = √(16 · 3) = 4√3.
3. Simplify √(9/16).
√9 / √16 = 3/4.
4. Simplify √(x²) for x ≥ 0.
The square root undoes the square: x.

✏️ Practice — try these, take hints as needed

1. Simplify √18.
Largest perfect-square factor of 18 is 9.
√(9·2).
—
3√2.
2. Simplify √75.
25 is a perfect-square factor.
√(25·3).
—
5√3.
3. Simplify √200.
100 · 2.
√100 · √2.
—
10√2.
4. Simplify √(25/49).
Split numerator and denominator roots.
√25/√49.
—
5/7.
5. Simplify √(36x²) for x ≥ 0.
√36 = 6 and √(x²) = x.
Multiply.
—
6x.

📝 Topic test — 8 questions

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