How Do You Simplify a Fraction with a Square Root?

Fractions that contain square roots appear often in algebra and geometry. Simplifying them usually involves rationalizing the denominator, rewriting so that no radical remains in the denominator. Here’s how to handle fractions with square roots.

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When the Square Root Is in the Denominator

Step 1: Multiply by a Form of 1

To rationalize a denominator like √5, multiply the fraction by √5/√5 (which equals 1). Example: 3/√5 × √5/√5 = 3√5/5. The denominator is now a rational number. For practice, check our free worksheets.

Step 2: Simplify the Numerator

If the numerator has a radical, simplify it first. For 2√12/√3, simplify √12 = 2√3 to get 4√3/√3, then rationalize: 4√3/√3 × √3/√3 = 4×3/3 = 4.

When the Square Root Is in the Numerator

If only the numerator has a radical, simplify the radical and reduce common factors. Example: √18/6 = 3√2/6 = √2/2. No rationalization needed unless your teacher requires the radical in a specific place.

When Both Numerator and Denominator Have Radicals

Rationalize by multiplying by the conjugate or by the denominator’s radical. For (√2 + 1)/√3, multiply by √3/√3 to get (√6 + √3)/3. Visit Effortless Math for more algebra resources.

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Why do we rationalize the denominator?

Convention and ease of computation. A rational denominator makes addition and comparison of fractions simpler.

What if the denominator is a sum like 1 + √2?

Multiply by the conjugate: (1 − √2)/(1 − √2). The product (1+√2)(1−√2) = 1 − 2 = −1, giving a rational denominator.