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

Rationalize a denominator by rewriting a fraction without a radical in the denominator.

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

Rationalizing Cube Roots and Denominators with Variables

With a cube root, multiplying by the same root is not enough, because ∛2 · ∛2 = ∛4, which is still a radical. Instead, multiply by whatever makes the radicand a perfect cube.

• 5/∛2: 2 needs two more factors of 2 to become 8, so multiply by ∛4/∛4. The denominator becomes ∛8 = 2, giving 5∛4/2.
• 3/∛9: 9 = 3², so one more 3 makes 27. Multiply by ∛3/∛3: 3∛3/∛27 = 3∛3/3 = ∛3.

The same idea works with variables. Assume the variables are positive.

• 6/√(2x): multiply by √(2x)/√(2x) to get 6√(2x)/(2x) = 3√(2x)/x.
• 4/(√x + 1): multiply by the conjugate √x − 1 to get 4(√x − 1)/(x − 1), with x ≠ 1.

Rationalizing the Numerator and Common Mistakes

Sometimes, especially in precalculus and calculus, you move the radical out of the numerator instead. The method is the same: multiply the top and bottom by the conjugate of the numerator.

Example: (√5 − √3)/2. Multiply by (√5 + √3)/(√5 + √3). The top becomes 5 − 3 = 2, so the result is 2/(2(√5 + √3)) = 1/(√5 + √3).

Watch for these mistakes:

1. Multiplying only the denominator. Multiply the top by the same thing, so the value stays the same.
2. Using the same binomial instead of the conjugate. (2 + √3)(2 + √3) = 7 + 4√3 still has a radical, but (2 + √3)(2 − √3) = 4 − 3 = 1.
3. Skipping the simplify step. 6/√12 = 6/(2√3) = 3/√3 = 3√3/3 = √3.
4. Canceling across a sum. In (6 + 2√5)/2 you divide both terms by 2 to get 3 + √5; you cannot cancel just one term.

Frequently Asked Questions

What is rationalizing a radical?

Rationalizing means rewriting a fraction so that no radical is left in the denominator, without changing its value. You multiply the top and bottom by a carefully chosen radical. For example, 6/√3 becomes 6√3/3 after multiplying by √3/√3, and that simplifies to 2√3. It is usually the last step before a radical answer counts as simplified.

How do you rationalize radical denominators?

If the denominator is one square root, multiply the top and bottom by that root. If it is a binomial such as 3 − √5, multiply by its conjugate, 3 + √5, so the middle terms cancel. Then 4/(3 − √5) = 4(3 + √5)/(9 − 5) = 4(3 + √5)/4 = 3 + √5. Always simplify any common factors at the end.

Is rationalizing the same as simplifying?

Not exactly. Simplifying a radical means pulling out perfect powers, such as √50 = 5√2. Rationalizing is one specific step that removes a radical from a denominator. A fully simplified answer usually needs both: 10/√8 = 10/(2√2) = 5/√2 = 5√2/2. Many teachers do not accept a final answer until it is rationalized.

Is a radical expression rational or irrational?

It depends on the number under the radical. √16 = 4 and ∛27 = 3 are rational, because the radicands are perfect powers. √5 and ∛2 are irrational, since their decimals never end or repeat. Rationalizing a denominator does not make the whole number rational: √2/2 is still irrational, but its denominator is now a whole number.

How do you rewrite radical expressions as rational exponents?

Use the rule ⁿ√(xᵐ) = x^(m/n): the index becomes the denominator and the power inside becomes the numerator. So √x = x^(1/2), ∛(x²) = x^(2/3) and ⁴√(x³) = x^(3/4). This form helps when rationalizing with variables, because 1/x^(1/3) needs x^(2/3) to make x^(3/3) = x in the denominator.

What grade is simplifying radical expressions?

Students usually meet square roots in 8th grade and learn to simplify radical expressions in Algebra 1, often in 9th grade. Rationalizing binomial denominators with conjugates and working with cube roots and rational exponents usually come in Algebra 2. The exact grade depends on the school’s course sequence.

Related Topics

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