Rationalizing the Denominator — How to & Examples

Q: What is the conjugate and when do you use it to rationalize?

The conjugate of a binomial expression $a + \sqrt{b}$ is $a - \sqrt{b}$ (you flip the sign between the two terms). You use the conjugate when the denominator is a sum or difference involving a radical, such as $3 + \sqrt{2}$. Multiplying by the conjugate creates a difference of squares, which eliminates the radical from the denominator.

Q: Does rationalizing the denominator change the value of the fraction?

No. You multiply the fraction by a form of 1 (such as $\frac{\sqrt{3}}{\sqrt{3}}$), which does not change the value. The original and rationalized fractions are equal — they are just written differently.

Rationalizing the Denominator

The process by which a fraction is rewritten so that the denominator contains only rational numbers. A variety of techniques for rationalizing the denominator are demonstrated below.

Square Roots (a > 0, b > 0, c > 0)	Examples

nth Roots (a > 0, b > 0, c > 0)	Examples


Example

Key Formula

\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b}

Where:

$a$ = The numerator of the original fraction (any real number)
$b$ = The positive number under the square root in the denominator
$\sqrt{b}$ = The irrational denominator to be eliminated

Worked Example

Problem: Rationalize the denominator of

\dfrac{5}{\sqrt{3}}

Step 1: Identify the radical in the denominator. Here the denominator is

\sqrt{3}

\frac{5}{\sqrt{3}}

Step 2: Multiply both the numerator and denominator by

\sqrt{3}

. This is equivalent to multiplying by 1, so the value of the fraction does not change.

\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}

Step 3: Multiply across. In the numerator:

5 \cdot \sqrt{3} = 5\sqrt{3}

. In the denominator:

\sqrt{3} \cdot \sqrt{3} = 3

\frac{5\sqrt{3}}{3}

Step 4: Check that the denominator is now rational. Since 3 is a rational number, the fraction is fully rationalized.

\frac{5\sqrt{3}}{3}

Answer:

\dfrac{5\sqrt{3}}{3}

Another Example

This example uses the conjugate method for a binomial denominator (sum of a rational number and a radical), whereas the first example handled a single square root in the denominator.

Problem: Rationalize the denominator of

\dfrac{3}{2 + \sqrt{5}}

Step 1: The denominator is a binomial containing a radical:

2 + \sqrt{5}

. To eliminate the radical, multiply by the conjugate

2 - \sqrt{5}

\frac{3}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}}

Step 2: Multiply the denominators using the difference of squares pattern:

(a + b)(a - b) = a^2 - b^2

(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1

Step 3: Multiply the numerator by distributing.

3(2 - \sqrt{5}) = 6 - 3\sqrt{5}

Step 4: Combine the results and simplify. Dividing by

-1

changes the signs.

\frac{6 - 3\sqrt{5}}{-1} = -6 + 3\sqrt{5} = 3\sqrt{5} - 6

Answer:

3\sqrt{5} - 6

Frequently Asked Questions

Why do you have to rationalize the denominator?

Rationalizing the denominator is a standard mathematical convention that makes expressions easier to compare, simplify, and compute with. Before calculators, dividing by a whole number was far simpler than dividing by an irrational number. Even today, rationalized forms are considered simplified form in most algebra courses, and many teachers require it.

What is the conjugate and when do you use it to rationalize?

The conjugate of a binomial expression

a + \sqrt{b}

a - \sqrt{b}

(you flip the sign between the two terms). You use the conjugate when the denominator is a sum or difference involving a radical, such as

3 + \sqrt{2}

. Multiplying by the conjugate creates a difference of squares, which eliminates the radical from the denominator.

Does rationalizing the denominator change the value of the fraction?

No. You multiply the fraction by a form of 1 (such as

\frac{\sqrt{3}}{\sqrt{3}}

), which does not change the value. The original and rationalized fractions are equal — they are just written differently.

Single radical denominator vs. Binomial radical denominator

	Single radical denominator	Binomial radical denominator
Example	$\dfrac{5}{\sqrt{3}}$	$\dfrac{3}{2 + \sqrt{5}}$
What to multiply by	The same radical: $\dfrac{\sqrt{3}}{\sqrt{3}}$	The conjugate: $\dfrac{2 - \sqrt{5}}{2 - \sqrt{5}}$
Key identity used	$\sqrt{a} \cdot \sqrt{a} = a$	$(a+b)(a-b) = a^2 - b^2$
Difficulty level	Straightforward — one multiplication step	Requires distribution and combining like terms

Why It Matters

Rationalizing the denominator appears frequently in Algebra 1, Algebra 2, and Precalculus whenever you simplify radical expressions. It is also essential in geometry (e.g., working with exact side lengths like

\frac{1}{\sqrt{2}}

in 45-45-90 triangles) and in calculus when simplifying limits. Standardized tests such as the SAT and ACT expect answers in rationalized form, so mastering this skill is important for both coursework and exams.

Common Mistakes

Mistake: Multiplying only the denominator by the radical, but forgetting to multiply the numerator by the same factor.

Correction: You must multiply both the numerator and the denominator by the same expression. This keeps the fraction equal to its original value because you are effectively multiplying by 1.

Mistake: Using the same binomial instead of the conjugate when the denominator has two terms (e.g., multiplying

2 + \sqrt{5}

2 + \sqrt{5}

instead of

2 - \sqrt{5}

Correction: Multiplying by the same binomial squares the denominator but does not eliminate the radical. You need the conjugate (flip the sign) so the difference of squares pattern removes the radical entirely.

Related Terms

Fraction — The type of expression being simplified
Denominator — The bottom of a fraction that is rationalized
Rational Numbers — The goal form for the denominator
Irrational Numbers — The type of number removed from the denominator
Square Root Rules — Rules used when multiplying radicals
Radical Rules — General rules for simplifying radical expressions
Factoring Rules — Used for difference of squares in conjugate method