Rationalize the Denominator

We rationalize the denominator to ensure that it becomes easier to perform any calculation on the rational number. When we rationalize the denominator in a fraction, then we are eliminating any radical expressions such as square roots and cube roots from the denominator. In this article, let's learn about rationalizing the denominator, its meaning, and methods with some examples.

Rationalizing is the process of multiplying a surd with another similar surd, to result in a rational number. The surd that is used to multiply is called the rationalizing factor (RF).

• To rationalize √x we need another √x: √x × √x = x.
• To rationalize a +√b we need a rationalizing factor a -√b: (a +√b) × (a -√b) = (a)² - (√b)² = a² - b.
• The rationalizing factor of 2√3 is √3: 2√3 × √3 = 2 × 3 = 6.

Rationalize the Denominator Meaning

Rationalizing the denominator means the process of moving a root, for instance, a cube root or a square root from the bottom of a fraction (denominator) to the top of the fraction (numerator). This way, we bring the fraction to its simplest form thereby, the denominator becomes rational.

The above-given table lists the irrational denominators and their equivalent rational values.

Before we learn how to rationalize a denominator, we need to know about conjugates. A conjugate is a similar surd but with a different sign. The conjugate of (7 + √5) is (7 - √5). In the process of rationalizing a denominator, the conjugate is the rationalizing factor. The process of rationalizing the denominator with its conjugate is as follows.

• Step 1: Multiply both the denominator and numerator by a suitable conjugate that will remove the radicals in the denominator.
• Step 2: We need to make sure that all the surds in the given fraction are in their simplified form.
• Step 3: If needed, we can simplify the fraction further.

Let us take an example of rationalizing the denominator of the fraction 1/(7+√5) to understand this concept better.

\(\begin{align} \frac{1}{7 + \sqrt 5} &= \frac{1}{7 + \sqrt 5} \times \frac{7 - \sqrt 5}{7 - \sqrt 5 } \\ &= \frac{7 - \sqrt 5}{(7)^2 - (\sqrt 5)^2} \\ &= \frac{7 - \sqrt 5}{49 - 5} \\ &= \frac{7 - \sqrt 5}{44} \end{align}\)

Another way to rationalize the denominator is to use algebraic identities. The algebraic formula used in the process of rationalization is (a² - b²) = (a + b)(a - b).

• For rationalizing (√a -√b), the rationalizing factor is (√a +√b).
• For rationalizing (√a + √b), the rationalizing factor is (√a − √b).
• (√a − √b) × (√a + √b) = (√a)² - (√b)² = a - b.

Let us understand this with an example. Consider the fraction 4/(√11 -√7). Let's rationalize the denominator in the following way:

\(\begin{align}\dfrac{4}{\sqrt 11 - \sqrt 7} &=\dfrac{4}{\sqrt 11 - \sqrt 7} \times \dfrac{\sqrt11 + \sqrt7}{\sqrt 11 + \sqrt 7}\\ &= \dfrac{4(\sqrt11 + \sqrt7)}{(\sqrt 11)^2 - (\sqrt 7)^2} \\&=\dfrac{4(\sqrt11 + \sqrt7)}{11 - 7} \\ &=\dfrac{4(\sqrt11 + \sqrt7)}{4} \\&= \sqrt11 + \sqrt7\end{align}\)

The same procedure that we followed to rationalize the denominator with 2 terms, we can follow those steps but with a little variation. Consider a denominator that has these three terms: a + b + c. We rationalized a denominator with 2 terms: a + b, by multiplying with its conjugate a - b. We can apply the same reasoning to rationalize a denominator that contains three terms by grouping the terms as a + b + c = (a + b) + c. As per the difference of squares formula, we have: [(a + b) + c] × [(a + b) - c] = (a + b)² − c².

Let's consider this example:

\(\begin{align}\dfrac{1}{1 + \sqrt 3 - \sqrt 5} &=\dfrac{1}{(1+\sqrt3) - \sqrt 5} \times \dfrac{(1+\sqrt3) + \sqrt5}{(1+\sqrt3) + \sqrt 5}\\ &= \dfrac{(1+\sqrt3) + \sqrt5}{(1+\sqrt 3)^2 - (\sqrt 5)^2} \\&=\dfrac{1+\sqrt3 + \sqrt5}{1+2\sqrt3 +3 -5} \\ &=\dfrac{1+\sqrt3 + \sqrt5}{2\sqrt3-1} \end{align}\)

Now, we can multiply the numerator and denominator with the conjugate of (2√3-1), which is (2√3+1).

\(\begin{align}\dfrac{1+\sqrt3 + \sqrt5}{2\sqrt 3 - 1} & =\dfrac{1+\sqrt3 + \sqrt5}{2\sqrt3 - 1} \times \dfrac{2\sqrt3 + 1}{2\sqrt3 + 1}\end{align}\)

= \(\begin{align}\dfrac{2\sqrt 3 + 2\sqrt 3 \sqrt 3 + 2\sqrt 3 \sqrt 5 + 1 + \sqrt 3 + \sqrt 5 }{(2\sqrt 3)^2 - (1)^2}\end{align}\)

= \(\begin{align}\dfrac{3\sqrt 3 + 6 + 2\sqrt 15 + 1 + \sqrt 5 }{12 - 1}\end{align}\)

= \(\begin{align}\dfrac{3\sqrt 3 + 7 + 2\sqrt 15 + \sqrt 5 }{11}\end{align}\)

Related Articles on Rationalize the Denominator

Check the following pages related to rationalize the denominator.

• Polynomials
• Rationalization
• Square Root
• Cube Root

Important Notes on Rationalize the Denominator:

Here is a list of a few points that should be remembered while studying about rationalize the denominator.

• The conjugate or the rationalizing factor of (√a +√b) is (√a -√b).
• The algebraic identity used in the process of rationalization is (a + b) × (a - b) = a² - b².

How do you Rationalize the Denominator and Simplify?

The process of rationalizing the denominator is as follows:

• Step 1: Multiply both the denominator and numerator by a suitable conjugate that will remove the radicals from the denominator.
• Step 2: We need to make sure that all the surds in the given fraction are in their simplified form.
• Step 3: If needed, we can simplify the fraction further.

How do you Rationalize a Denominator With 2 Terms?

Here are the steps to be followed to rationalize the denominator with 2 terms:

• To rationalize the denominator with two terms, we multiply the numerator and denominator of the fraction with the conjugate of the denominator.
• To find the conjugate of two terms we have to change the sign between the two terms.
• Combine all the like terms and simplify the radicals.
• Try to reduce the fraction to its simplest form, if possible.

How do you Rationalize the Denominator of 1/(2+√3)?

Let's rationalize the denominator of the given fraction 1/(2+√3) by multiplying both the numerator and the denominator with its conjugate (2 - √3).

\(\begin{align} \dfrac{1}{2 + \sqrt3} &= \dfrac{1}{2 + \sqrt3} \times \dfrac{2 - \sqrt3}{2 - \sqrt3} \\& = \dfrac{2 - \sqrt3}{(2)^2 - (\sqrt3)^2} \\ &= \dfrac{2 - \sqrt3}{4 - 3} \\&=\dfrac{2 - \sqrt3}{1} \\ &= 2 - \sqrt3\end{align} \)

How do you Rationalise a Denominator With 3 Terms?

The same procedure that we followed to rationalize the denominator with 2 terms, we can follow those steps but with a little variation. Consider a denominator that has these three terms: a + b + c. We rationalized a denominator with 2 terms: a + b, by multiplying with its conjugate a - b. We can apply the same reasoning to rationalize a denominator that contains three terms by grouping the terms as a + b + c = (a + b) + c. As per the difference of squares formula, we have: [(a + b) + c] × [(a + b) - c] = (a + b)² − c².

How do you Rationalize the Denominator of 1/√7?

To rationalize √7 in the denominator, we require another √7.

\(\frac{1}{\sqrt{7}}=\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}\)

= √7 /(√7 × √7) = √7/7.

Therefore the answer is √7/7.

How do you Rationalize the Denominator With a Square Root in It?

To rationalize a denominator with a square root, we multiply both the numerator and the denominator with the same square root. For example, to rationalize the denominator 1/√5, we will multiply the numerator and denominator with √5.

\(\frac{1}{\sqrt{5}}=\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\)

√5/(√5 × √5) = √5/5.

What does Rationalise the Denominator Mean?

Rationalizing the denominator means the process of moving a root, for instance, a cube root or a square root from the bottom of a fraction to the top of the fraction. This way, we bring the fraction to its simplest form thereby, the denominator becomes rational.