How to Rationalize Radical Expressions? (+FREE Worksheet!)

Rationalizing radical expressions means rewriting a fraction so that no radical appears in the denominator. This is a standard requirement for a simplified answer in algebra, and the technique depends on whether the denominator is a single radical term (monomial) or a binomial containing a radical. This guide covers both cases with clear steps, four worked examples, two video lessons, and practice problems.

What Does Rationalizing the Denominator Mean?

A fraction like \(\color{blue}{\frac{1}{\sqrt{2}}}\) has an irrational denominator — a denominator that is not a rational number. Rationalizing means multiplying the fraction by a carefully chosen form of 1 so that the denominator becomes rational (a whole number or simple fraction), while the value of the expression does not change.

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Case 1: Monomial Denominator

When the denominator is a single radical such as \(\color{blue}{\sqrt{a}}\), multiply top and bottom by that same radical.

\(\color{blue}{\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}}\)

This works because \(\color{blue}{\sqrt{a} \times \sqrt{a} = a}\).

• \(\color{blue}{\frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}}\)
• \(\color{blue}{\frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}}\)

Case 2: Binomial Denominator with a Radical

When the denominator is a binomial such as \(\color{blue}{(a + \sqrt{b})}\), multiply by its conjugate \(\color{blue}{(a – \sqrt{b})}\). The conjugate differs only in the sign between the terms. The product of conjugates eliminates the radical using the difference of squares:

\(\color{blue}{(a + \sqrt{b})(a – \sqrt{b}) = a^{2} – b}\)

• \(\color{blue}{\frac{5}{(2 + \sqrt{3})} \times \frac{(2 – \sqrt{3})}{(2 – \sqrt{3})} = \frac{5(2 – \sqrt{3})}{(4 – 3)} = \frac{5(2 – \sqrt{3})}{1} = 10 – 5\sqrt{3}}\)

Step-by-Step Summary

1. Identify whether the denominator is a monomial radical or a binomial containing a radical.
2. Monomial: multiply numerator and denominator by the same radical.
3. Binomial: multiply numerator and denominator by the conjugate of the denominator.
4. Multiply out numerator and denominator.
5. Simplify the result by reducing any common factors.

Watch: Rationalizing the Denominator (Video Lesson)

Khan Academy explains the monomial case clearly with a step-by-step Algebra I example:

Rationalizing Radical Expressions — Worked Examples

Example 1: Rationalize \(\color{blue}{\frac{1}{\sqrt{2}}}\).

Multiply by \(\color{blue}{\frac{\sqrt{2}}{\sqrt{2}}}\): \(\color{blue}{\frac{\sqrt{2}}{(\sqrt{2} \times \sqrt{2})} = \frac{\sqrt{2}}{2}}\).
Answer: \(\color{blue}{\frac{\sqrt{2}}{2}}\)

Example 2: Rationalize \(\color{blue}{\frac{3}{\sqrt{5}}}\).

Multiply by \(\color{blue}{\frac{\sqrt{5}}{\sqrt{5}}}\): \(\color{blue}{\frac{3\sqrt{5}}{5}}\).
Answer: \(\color{blue}{\frac{3\sqrt{5}}{5}}\)

Example 3: Rationalize \(\color{blue}{\frac{5}{(2 + \sqrt{3})}}\).

Conjugate: \(\color{blue}{(2 – \sqrt{3})}\). Multiply: \(\color{blue}{\frac{5(2 – \sqrt{3})}{((2 + \sqrt{3})(2 – \sqrt{3}))} = \frac{5(2 – \sqrt{3})}{(4 – 3)} = \frac{5(2 – \sqrt{3})}{1}}\).
Answer: \(\color{blue}{10 – 5\sqrt{3}}\)

Example 4: Rationalize \(\color{blue}{\frac{2}{(\sqrt{7} – 1)}}\).

Conjugate: \(\color{blue}{(\sqrt{7} + 1)}\). Multiply: \(\color{blue}{\frac{2(\sqrt{7} + 1)}{(7 – 1)} = \frac{2(\sqrt{7} + 1)}{6} = \frac{(\sqrt{7} + 1)}{3}}\).
Answer: \(\color{blue}{\frac{(\sqrt{7} + 1)}{3}}\)

Algebra II: Rationalizing Expressions Video

This Khan Academy lesson covers more complex rationalizing situations including binomial denominators:

Exercises for Rationalizing Radical Expressions

Rationalize the denominator and simplify.

1. \(\color{blue}{\frac{4}{\sqrt{3}}}\)
2. \(\color{blue}{\frac{6}{\sqrt{6}}}\)
3. \(\color{blue}{\frac{1}{(1 + \sqrt{2})}}\)
4. \(\color{blue}{\frac{10}{\sqrt{5}}}\)
5. \(\color{blue}{\frac{3}{(\sqrt{5} + \sqrt{2})}}\)

Answers

1. \(\color{blue}{\frac{4\sqrt{3}}{3}}\)
2. \(\color{blue}{\sqrt{6}}\)  (since \(\color{blue}{\frac{6\sqrt{6}}{6} = \sqrt{6}}\))
3. \(\color{blue}{\sqrt{2} – 1}\)  (conjugate: \(\color{blue}{1 – \sqrt{2}}\); numerator: \(\color{blue}{1 – \sqrt{2}}\); denominator: \(\color{blue}{1 – 2 = -1}\); final: \(\color{blue}{\sqrt{2} – 1}\))
4. \(\color{blue}{2\sqrt{5}}\)  (since \(\color{blue}{\frac{10\sqrt{5}}{5} = 2\sqrt{5}}\))
5. \(\color{blue}{\sqrt{5} – \sqrt{2}}\)  (conjugate: \(\color{blue}{\sqrt{5} – \sqrt{2}}\); denominator: \(\color{blue}{5 – 2 = 3}\); simplifies to \(\color{blue}{\sqrt{5} – \sqrt{2}}\))

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Want More Practice?

We haven’t published a worksheet built specifically for Rationalizing Radical Expressions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Graphing Square Root Functions Worksheet
• Download Properties of Exponents Worksheet

Frequently Asked Questions

Why must we rationalize the denominator?

In simplified form, denominators should be rational numbers. Irrational denominators make comparison, computation, and further algebra harder. Rationalizing produces a cleaner, standard form.

What is a conjugate?

The conjugate of \(\color{blue}{(a + b)}\) is \(\color{blue}{(a – b)}\). Multiplying by the conjugate uses the difference of squares: \(\color{blue}{(a + b)(a – b) = a^{2} – b^{2}}\). When b contains a square root, \(\color{blue}{b^{2}}\) becomes rational, eliminating the radical from the denominator.

Does rationalizing change the value of the expression?

No. You are multiplying by a form of 1 (conjugate over itself, or radical over itself), so the value is unchanged. Only the form changes — the denominator becomes rational.

Related Topics

• Simplifying Radical Expressions
• Multiplying Radical Expressions
• Adding and Subtracting Radical Expressions
• Radical Equations
• Domain and Range of Radical Functions