Rationalize the denominator calculator is an online tool that gives the rationalized denominator for any input. STUDYQUERIES’s online rationalize denominator calculator makes calculations faster and easier. The result is displayed in a fraction of a second.

How to Use Rationalize the Denominator Calculator?

To rationalize the denominator calculator, follow these steps:

  • Step 1: Enter the numerator and denominator values in the input fields
  • Step 2: Click the “Rationalize Denominator” button to get the results
  • Step 3: The output field displays the result

Rationalize The Denominator Calculator

What Is Rationalization?: Step By Step Guide

In elementary algebra, rationalization is used to eliminate irrational numbers from denominators. To rationalize the denominator, many rationalizing techniques are used. By definition, rationalizing means maximizing the efficiency of something. In mathematics, its adoption means reducing the equation into a more efficient and simple form.

Rationalize The Denominator Calculator
Rationalize The Denominator Calculator

In other words, we can say that the rationalization process is used to remove a radical or an imaginary number from the denominator of an algebraic fraction. That is, remove the radicals from a fraction so that the denominator contains only rational numbers. In this section, let’s review a few terms related to rationalization.

Radical

Radical expressions are expressions that use a root, such as a square root or cube root. A radical expression looks like this:

$$\sqrt{a+b}\ is\ radical$$

Radicand

We are finding the root of the term Radicand.

Radicand
Radicand

The radicand in the above expression, for example, is (a + b).

Radical Symbol

The $$\sqrt{}$$ symbol means “root of”. It is important to consider the length of the horizontal bar. Variables or constants that are a part of the root function are represented by the length of the bar.

Radical Symbol
Radical Symbol

Variables or constants not under the root symbol are therefore not part of the root.

Degree

In the figure below, the degree is represented by a number. 2 represents square roots, 3 represents cube roots. Furthermore, they are referred to as 4th root, 5th root, and so on.

Degree
Degree

When this is not mentioned, we take it as square root by default.

Conjugate

Any math conjugate of a binomial is another exact binomial with the opposite sign between its two terms. The conjugate of (x + y) is (x – y), and vice versa. Both of these binomials are conjugates of each other.

Rationalizing a Surd

Surds are irrational numbers that cannot be further simplified in their radical form. For example, an irrational number $$\sqrt{8}$$ can be simplified further as $$2\sqrt{2}$$, whereas $$\sqrt{2}$$ cannot be simplified any further. Thus $$\sqrt{2}$$ is a surd.

Examples of a monomial radical: $$\sqrt{6}, 3\sqrt{2},\sqrt[3]{2}$$

Examples of a binomial radical: $$\sqrt{3} + \sqrt{6}, 1 – \sqrt{2}$$

The procedure for rationalizing an expression depends on whether the radical is monomial or binomial.

Rationalizing a Monomial Radical

A radical or surd rationalized in the denominator follows different steps depending on the degree of the polynomial or if the radical is a monomial or polynomial. Monomials are polynomials with only one term.

$$\sqrt{2}, \sqrt{7}x,\sqrt[3]{7x}$$ etc. could be in the denominators.

Method:

  •  Suppose the denominator contains a radical, as in this fraction: $$\frac{a}{\sqrt{b}}$$. Here, the radical must be multiplied and divided by $$\sqrt{b}$$ and further simplified.
  • For a polynomial with a monomial radical in the denominator, say of the form,$$a\sqrt[m]{x^n}$$ such that n < m, the fraction must be multiplied by a quotient containing $$a\sqrt[m]{x^{m-n}}$$ both in the numerator and denominator. This gives us the result $$a\sqrt[m]{x^m}$$ which can be replaced by x and hence free of the radical term.

Let us go through this technique step by step using the following examples.

Let us rationalize the fraction: $$\frac{2}{\sqrt{7}}$$

Step 1: Analyze the fraction – The given fraction has a monomial radical $$\sqrt{7}$$ in the denominator that should be rationalized. It is not necessary to consider the numerator while examining the fraction or simplifying it since it can have a radical.

Rationalizing A Monomial Radical Example
Rationalizing A Monomial Radical Example

Step2: By multiplying the fractions in the denominator and numerator $$\sqrt{7}$$

$$\frac{2\times \sqrt{7}}{\sqrt{7}\times \sqrt{7}}$$

Step3: Simplify the expression as needed.

$$\frac{2\times \sqrt{7}}{\sqrt{7}\times \sqrt{7}}=\frac{2\sqrt{7}}{7}$$

Rationalizing a Binomial Radical

If the denominator has a radical expression of the form $$a+\sqrt{b}\ or\ a+i\sqrt{b}$$ the fraction must be multiplied by the conjugate of the expression i.e., $$a-\sqrt{b}\ or\ a-i\sqrt{b}$$ both in the numerator and denominator.

Method: Suppose the denominator contains a radical expression, $$a+\sqrt{b}\ or\ a+i\sqrt{b}$$ Here, the radical must be multiplied by its conjugate.

Let us have a look at the following fraction, $$\frac{5}{2-\sqrt{3}}$$

The denominator needs to be rationalized.

$$\frac{5}{2-\sqrt{3}}\times \frac{2+\sqrt{3}}{2+\sqrt{3}}$$

This is further simplified and evaluated as $$5(2+\sqrt{3})$$

Using algebraic identities, the denominator is further expanded.

Let us learn the technique to rationalize the following fraction: $$\frac{\sqrt{7}}{2+\sqrt{7}}$$

Step1: Examine the fraction – The denominator of the above fraction has a binomial radical i.e., is the sum of two terms, one of which is an irrational number.

Step2: Multiply the numerator and denominator of the fraction with the conjugate of the radical.

$$\frac{\sqrt{7}}{2+\sqrt{7}}\times \frac{2-\sqrt{7}}{2-\sqrt{7}}$$

Step3: Simplify the expression as needed.

$$\frac{\sqrt{7}}{2+\sqrt{7}}\times \frac{2-\sqrt{7}}{2-\sqrt{7}}$$

Using difference of squares algebraic property, $$(a + b)(a – b) = a^2 – b^2$$

$$\frac{\sqrt{7}}{2+\sqrt{7}}\times \frac{2-\sqrt{7}}{2-\sqrt{7}}=\frac{2\sqrt{7}-\sqrt{7}\times \sqrt{7}}{2^2-{\sqrt{7}}^2}$$

$$=\frac{2\sqrt{7}-7}{4-7}$$

$$=\frac{7-2\sqrt{7}}{3}$$

Rationalizing a Cube Root

This method can be generalized to the roots of any order while rationalizing. For the given fraction with cube root radical in the denominator, $$\frac{1}{\sqrt[3]{5}}$$ we can follow the given steps,

Step1: Examine the fraction – The fraction has a radical in the form of a cube root in the denominator.

$$\frac{1}{\sqrt[3]{5}}$$

Step2: Multiply the numerator and denominator of the fraction by a factor that makes the exponent of the denominator 1. In this case, that factor would be $$5^{\frac{2}{3}}$$

Step3: Simplify the expression as needed.

$$\frac{1}{\sqrt[3]{5}}\times \frac{5^{\frac{2}{3}}}{5^{\frac{2}{3}}}$$

$$=\frac{5^{\frac{2}{3}}}{{5^{\frac{1}{3}}}\times {5^{\frac{2}{3}}}}$$

$$=\frac{5^{\frac{2}{3}}}{5^{\frac{1}{3}+\frac{2}{3}}}$$

$$=\frac{5^{\frac{2}{3}}}{5}$$

$$=\frac{\sqrt[3]{25}}{5}$$

How to Rationalize Denominator?

When you look at the definition of “rationalize”, it will become clearer as to what exactly rationalizing a denominator means. The numbers like 1/2, 5, and 0.25 are all rational numbers i.e., they can be expressed as a ratio of two integers like 1/2,5/1,1/4 respectively.

Alternatively, some radicals cannot be expressed as the ratio of two integers since they are irrational numbers. Hence, the denominator of the expression must be rationalized so that it becomes a rational number. Here is a table with the equivalent rational values of irrational numbers.

The process of rationalizing the denominator consists of moving a root, such as a cube root or a square root, from the bottom of a fraction (denominator) to the top (numerator). Thus, the fraction is reduced to its simplest form and the denominator becomes rational.

Rationalize the Denominator
Rationalize the Denominator

The table above lists the irrational denominators and their rational equivalents.

Rationalize the Denominator Using Conjugates

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 $$(7 + \sqrt{5})\ is\ (7 – \sqrt{5})$$ 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 the denominator and numerator by a suitable conjugate that removes the radicals from the denominator.
  • Step 2: Make sure that all the numbers in the given fraction are simplified.
  • Step 3: We can simplify the fraction further if necessary.

Here is an example of rationalizing the denominator of the fraction $$\frac{1}{7+√5}$$ to understand this concept better.

$$\frac{1}{7+√5}=\frac{1}{7+√5}\times \frac{7-√5}{7-√5}$$

$$=\frac{7-√5}{(7)^2-(\sqrt{5})^2}$$

$$=\frac{7-√5}{49-5}$$

$$=\frac{7-√5}{44}$$

Using Algebraic Identities, Rationalize The Denominator

Another way to rationalize the denominator is to use algebraic identities. The algebraic formula used in the process of rationalization is $$(a^2 – b^2) = (a + b)(a – b)$$

  • $$For\ rationalizing\ (\sqrt{a} -\sqrt{b})\ the\ rationalizing\ factor\ is\ (\sqrt{a} +\sqrt{b})$$
  • $$For\ rationalizing\ (\sqrt{a} + \sqrt{b})\ the\ rationalizing\ factor\ is\ (\sqrt{a} − \sqrt{b})$$
  • $$(\sqrt{a} − \sqrt{b})\times (\sqrt{a} + \sqrt{b}) = (\sqrt{a})^2 – (\sqrt{b})^2 = a – b$$

Let us understand this with an example. Consider the fraction \frac{4}{\sqrt{11} -\sqrt{7}}. Let’s rationalize the denominator in the following way:

$$\frac{4}{\sqrt{11} -\sqrt{7}}=\frac{4}{\sqrt{11} -\sqrt{7}}\times \frac{\sqrt{11} +\sqrt{7}}{\sqrt{11} +\sqrt{7}}$$

$$=\frac{4(\sqrt{11} +\sqrt{7})}{(\sqrt{11})^2 +(\sqrt{7})^2}$$

$$=\frac{4(\sqrt{11} +\sqrt{7})}{11-7}$$

$$=\frac{4(\sqrt{11} +\sqrt{7})}{4}$$

$$=\sqrt{11} +\sqrt{7}$$

Rationalize the Denominator With 3 Terms

We can follow the same steps we used to rationalize the denominator with two terms, but with a few variations. Consider a denominator with three terms: a + b + c. We rationalized a denominator with 2 terms: a + b, by multiplying with its conjugate a – b. In the same way, we can rationalize a denominator that contains three terms by grouping them as a + b + c = (a + b) + c. Using the difference of squares formula, we get: $$[(a + b) + c] \times [(a + b) – c] = (a + b)^2 − c^2$$

Let’s consider this example

$$\frac{1}{1+\sqrt{3}-\sqrt{5}}=\frac{1}{(1+\sqrt{3})-\sqrt{5}}\times \frac{1+\sqrt{3})+\sqrt{5}}{(1+\sqrt{3})+\sqrt{5}}$$

$$=\frac{1+\sqrt{3})+\sqrt{5}}{(1+\sqrt{3})^2+(\sqrt{5})^2}$$

$$=\frac{1+\sqrt{3}+\sqrt{5}}{1+2\sqrt{3}+3-5}$$

$$=\frac{1+\sqrt{3}+\sqrt{5}}{2\sqrt{3}-1}$$

Now, we can multiply the numerator and denominator with the conjugate of $$(2\sqrt{3}-1)\ which\ is\ (2\sqrt{3}+1)$$

$$\frac{1+\sqrt{3}+\sqrt{5}}{2\sqrt{3}-1}\times \frac{2\sqrt{3}+1}{2\sqrt{3}+1}=\frac{1+\sqrt{3}+\sqrt{5}}{2\sqrt{3}-1}\times \frac{2\sqrt{3}+1}{2\sqrt{3}+1}$$

$$=\frac{2\sqrt{3}+2\sqrt{3}\sqrt{3}+2\sqrt{3}\sqrt{5}+1+\sqrt{3}+\sqrt{5}}{(2\sqrt{3})^2-(1)^2}$$

$$=\frac{3\sqrt{3}+6+2\sqrt{15}+1+\sqrt{5}}{12-1}$$

$$=\frac{3\sqrt{3}+7+2\sqrt{15}+\sqrt{5}}{11}$$

Important Points To Remember

The following points should be kept in mind while studying rationalization:

  • Rationalization is the process of removing a radical or imaginary number from the numerator of an algebraic fraction. In other words, remove the radicals in a fraction so that the denominator only contains a rational number.
  • A radical is an expression that uses a root, such as the square root or cube root. For example, an expression of the form: √(a + b) is radical.
  • Math conjugates of binomials represent another exact binomial with the opposite sign between its two terms

FAQs

How do you do rationalization?

  • Step1: Examine the fraction – The fraction has a radical in the form of a cube root in the denominator.
  • Step2: Multiply the numerator and denominator of the fraction by a factor that makes the exponent of the denominator 1.
  • Step3: Simplify the expression as needed.

How do you rationalize the denominator?

To rationalize the denominator, you must multiply both the numerator and the denominator by the conjugate of the denominator. Remember to find the conjugate all you have to do is change the sign between the two terms. So this is the required rationalization of the given expression.

Why is Pi not a surd?

Only the square roots of square numbers are rational. Similarly Pi (π) is an irrational number because it cannot be expressed as a fraction of two whole numbers and it has no accurate decimal equivalent.

Is root 12 a pure surd?

Pure surd has no rational factor except unity. For example, 2√2, 2√5,2√7, 2√12, 3√15, 5√30, 7√50, n√x all are pure surds as these have rational numbers only under the radical sign or the whole expression purely belongs to a surd.

Can Surds have decimals?

A surd is an expression that includes a square root, cube root, or another root symbol. Surds are used to write irrational numbers precisely – because the decimals of irrational numbers do not terminate or recur, they cannot be written exactly in decimal form.