Rationalize the denominator: Meaning, Examples, Problems and More

We rationalize the denominator so that any computation on the rational number becomes easy. When we rationalize the denominator in a fraction, we remove any radical expressions from the denominator, such as square roots and cube roots. When dealing with radical expressions, we use the rationalization approach. Assume we can ‘rationalize’ the denominator to transform it to a rational number. We need IDs made out of square roots for this. So, in this article, we are talking about this topic. So, keep reading to know more about it.

Rationalize the denominator meaning

When a given fraction has a radical word or a surd in the denominator, we rationalize the denominator. Square root and cube root are two examples of radical words. If the denominator of a two-term mathematical statement contains a radical, we must multiply both the numerator and the denominator by the denominator’s conjugate. This is known as rationalization.

Read Also:Force Equation – Types, Formulae

In other words, rationalizing the denominator involves transferring the radical term (square root or cube root) to the numerator, resulting in a denominator that is a whole integer. When we rationalize the denominator, finding the sum or difference of given fractions becomes simple. For example, 2/√2 is an irrational denominator fraction. When we rationalize it, it becomes √2. As a result, the denominator is a whole number, namely 1. With examples, let us learn how to rationalize the denominator in this post.

Rationalize the denominator calculator

The rationalized denominator calculator is a free online application that calculates the rationalized denominator for a given input. Any online rationalization of the denominator calculator tool speeds up and simplifies the calculations, displaying the answer in a fraction of a second.

The following is the process for rationalizing the denominator calculator:

  • In the input area, enter the numerator and denominator values.
  • To obtain the output, click the “Rationalize Denominator” button.
  • The output field will reveal the result.

Rationalization is the process of eliminating imaginary numbers from an algebraic expression’s denominator. It is the process of transferring the radical (i.e., square root or cube root) from the fraction’s denominator to its denominator (numerator). To eliminate the radicals, multiply the numerator and denominator by the denominator’s conjugate.

Rationalize the denominator examples

Example 1

Simplify 1/√252

Solution: Given, 1/√252

Prime factorisation of 252 = 2 x 2 x 3 x 3 x 7

= 1/√( 2 x 2 x 3 x 3 x 7)

Removing the square values from the root

= 1/[2 x 3√7]

= 1/6√7

To rationalize, multiply and divide by 7.

= 1/6√7 x (√7 x √7)

= √7/(6 x 7)

= √7/42

Example 2

Simplify: 3√10-5√6 / 4√10+2√6

Solution: Given, 3√10-5√6 / 4√10+2√6

Numerator and denominator are multiplied by the denominator’s conjugate.

3√10-5√6 / 4√10+2√6 . 4√10-2√6 / 4√10-2√6

Multiply the terms

12√100-6√60-20√60+10√36 / 16√100-8√60+8√60-4√36

Simplify similar phrases.

=12(10)-26(2√15)+10(6) / 16(10)-4(6)

=120-52√15+60 / 160-24

=180-52√15 / 136

=45-13√15 / 34

Example 3

Justify the denominator 5/2. If necessary, simplify even more.

The square root of 2 is a radical statement in the denominator. Remove the radical at the bottom by multiplying by itself, which is 2 since 2 * 2 = 4 = 2.

However, we modify the “meaning” or worth of the original fraction by doing so. To balance it out, repeat the process on top by multiplying by the same number. We are simply multiplying the original fraction by 2/2, which is identical to 1. Remember that multiplying any number by one returns the original number; so, we modify the shape but not the original meaning of the number! So the straightforward answer to this problem is presented below.

5/ √2 * √2/ √2

= 5√2/ √4

=5√2 / 2

Because the final solution lacks a radical symbol, we may argue that we have effectively justified it.

Rationalize the denominator problems

Problem 1

Rationalize 1/√11.

Solution:

Because the supplied fraction contains an irrational denominator, we must rationalize it and simplify it. To explain, multiply the numerator and denominator of the given fraction by root 11, i.e., 11.

1/√11 * √11/√11

=√11/11

As a result, the necessary rationalized form of the provided denominator is: √11/11.

Problem 2

Rationalize 1/√21.

Solution:

The denominator of the given fraction is irrational. As a result, we must simplify the situation by rationalizing the stated denominator. To do this, we must multiply and divide the supplied fraction by root 21, i.e., 21. So, 1/√21 * √21/√21 = √21/21. As a result, the needed rationalized fraction is: √21/21.

Problem 3

Rationalize 1/√39.

Solution:

Because the provided fraction contains an irrational denominator. To make the computations easier, we need to simplify them, which means rationalizing the denominator. To accomplish so, we must multiply the fraction’s numerator and denominator by root 39, i.e., 39. So, 1/√39 * √39/√39 = √39/39. As a result, the needed rationalized fraction is: √39/39.

Rationalize the denominator and simplify

When dealing with a binomial denominator incorporating Radicals, the identities listed below can be used. We may simplify the terms in the numerator and denominator by using these identities.

ExpressionConjugateProduct
√a + b√a – b(√a + b)(√a – b) = a – b2
√a – b√a + b(√a – b)(√a + b) = a – b2
a + √ba – √b(a + √b)(a – √b) = a2 – b
a – √ba + √b(a – √b)(a + √b) = a2 – b
√a + √b√a – √b(√a + √b)(√a – √b) = a – b
√a – √b√a + √b(√a – √b)(√a + √b) = a – b

Rationalize the denominator 1/3+√2

We obtain after rationalizing 3/7 –  3/7

Explanation in detail:

Given: 1/ 3+2

We must justify the provided expression’s denominator.

Consider, 1/ 3+2

=  1/ 3+2 * 3-2 / 3-2

= 3-√3 / (3)^2-(√2)^2

=3-√3 / 9-2

=3-√3 / 7

=3/7-√3/7

As a result of our reasoning, we arrive at 3/7-√3/7.

Rationalize the denominator 1/√7

Rationalize 1/√7

Solution:

Dividing and multiplying by √7, we get 1/√7 = (1/√7) × (√7/√7) = √7/7.

Rationalize the denominator with square roots

We dislike radicals in the numerator. On the one hand, if we observe 2√3, it’s simply two square roots of three, which (at least to mathematicians) sounds and appears OK. On the other hand, 2 / √3 is an abomination that no self-respecting scientist can accept. We say, let them have their quirks.

The definition of rationalization is to please those picky mathematicians. In arithmetic, rationalization refers to the process of transferring radicals from the denominator to the numerator, or rationalizing the denominator of an equation. The overall value will most likely remain irrational; it’s only that the amount beneath the line will not.

Theoretically, rationalization in arithmetic reduces down to multiplying the statement by 1. Yes, by the same number that makes no difference in multiplication. The difficulty, though, is to have this 1 worded in such a manner that it truly makes a difference. Following all, 1 = 2 / 2 = 2020 / 2020 = √13 / √13 = (9 – ³√7) / (9 – ³√7).

To put it another way, we represent 1 as a reasonable quotient. Then we use the concept that multiplying fractions equals numerator times numerator over denominator times denominator. For example, suppose we have an expression of the type a / √b, and we want to know how to get rid of a square root in that situation. We could write: a / √b = (a / √b) * 1 = (a / √b) * (b / √b).

Next step

We decided to write 1 as b / b. We can, for several reasons. This results in: a / √b = (a / √b) * (√b / √b) = (a * √b) / (√b * √b) = (a√b) / b = (a/b) * √b.

Finally, we arrived at an identical phrase, but the root is no longer in the denominator. To be more specific, the definition of rationalization in this context was to convert what we had into a fraction (i.e., a rational number) times the same radical that caused the difficulty in the first place.

Rationalize the denominator with one terms

When you analyze the meaning of “rationalize,” the concept of rationalizing a denominator becomes clearer. Remember that the numbers 5,1/2 and 0.75vare all rational numbers since they may be written as a ratio of two integers (5/1,½ and 3/4, and respectively). Irrational numbers are those that cannot be expressed as a ratio of two integers. As a result, rationalizing a denominator is changing the phrase so that the denominator becomes a rational number.

Let us begin with the fraction 1/√2. It has an irrational number as its denominator √2. This makes determining what the value of 1/√2 is challenging.

If you double this fraction by one, you may rename it without altering its value. Set 1 equal to √2/√2 in this example. Take note of what occurs.

1/√2 *1= 1/√2*√2/√2 = √2 / √2*√2 = √2 / √4 = √2/ 2.

The new fraction’s denominator is no longer a radical (notice, however, that the numerator is). So, why do we multiply 1/√2 by √2/√2? You already knew that the square root of an integer multiplied by itself is a whole number.

Rationalize the denominator with two terms

The steps for doing rationalization on denominators with two terms are listed below.

  • Multiply both the numerator and the denominator by the conjugate of the denominator.
  • Distribute the numerator and denominator or use the FOIL approach.
  • We may multiply numbers within a radical by numbers within a radical and numbers outside a radical by numbers outside a radical. Then, mix similar phrases.
  • Now combine the acquired similar words and simplify the radical terms.
  • If feasible, transform the fraction to a simpler form.
  • For the outcome of the preceding step, we may need to lower each integer outside the radical by the same number.
  • We cannot lower the fraction if we cannot reduce each number outside the radical by the same number. Let’s look at an example to assist you understand.

Example

Justify the denominator of 2/√8-√7

Solution:

The following expression is: 2/√8-√7.

Because the conjugate of √a – √b is √a + √b, the conjugate of √8 – √7 is √8 + √7. When we multiply the numerator and denominator by √8 + √7, we obtain;

2/√8-√7 * √8+√7 / √8+√7

Using the FOIL approach and appropriate IDs,

2/√8+√7 / (√8)^2+(√7)^2

In terms of simplicity,

=2/√8+√7 / (8-7)

=2/√8+√7 / 1

=2/√8+√7

As a result, the denominator found here is reasonable.

Rationalize the denominator with three terms

We may use the same procedures we used to rationalize the denominator with two terms, but with a slight difference. Consider a denominator with the following three terms: a + b + c. By multiplying with its conjugate, a – b, we rationalize a denominator with two terms: a + b. We may use the same logic to rationalize a denominator with three terms by combining them as a + b + c = (a + b) + c. The difference of squares formula gives us: [(a + b) + c] × [(a + b) – c] = (a + b)2 − c2.

Consider the following example:

1/1+√3−√5= 1/(1+√3)−√5 × (1+√3)+√5 / (1+√3)+√5

=(1+√3)+√5 / (1+√3)^2−(√5)^2

=1+√3+√5 / 1+2√3+3−5

=1+√3+√5 / 2√3−1

We may now multiply the numerator and denominator by the conjugate of (2√3-1), which is (2√3+1).

1+√3+√5 / 2√3−1

=1+√3+√5 / 2√3−1 × 2√3+1 / 2√3+1

= 2√3+2√3√3+2√3√5+1+√3+√5 / (2√3)^2 − (1)^2

= 3√3+6+2√15+1+√5 / 12−1

= 3√3+7+2√15+√5 / 11

Rationalize the denominator with conjugates

Before we can learn how to justify a denominator, we must first understand conjugates. A conjugate is a surd that is comparable but has a different sign. (7 + √5) has the conjugate (7 – √5). The conjugate is the rationalizing factor in the process of rationalizing a denominator. The following is the procedure for rationalizing the denominator with its conjugate.

  • Step 1: Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.
  • Step 2: We must ensure that all of the surds in the specified fraction are simplified.
  • Step 3: If necessary, we can simplify the fraction even further.

To grasp this notion, consider rationalizing the denominator of the fraction 1/(7+√5) better.

1/7+√5=1/7+√5 × 7−√5 / 7−√5

=7−√5 / (7)^2−(√5)^2

=7−√5 / 49−5

=7−√5 / 44

Rationalize the denominator with cube root

If you have a cube root, b=a√3, then b√3=a by definition. To get the radicand, multiply three copies of the cube root (the a value under the radix).

If you were given 3√a^3 (like 3√8), the cube root is a (=2) since a^3 (=2^3) is the radicand. This is crucial to comprehending how to justify a denominator.

Take a look at 1/3√a.

You have one copy of an under the radical; you must have two additional products under the radical to obtain the cube root of a perfect cube, which is the same as a. So multiply 3√a *3√a^2 by 3 and simplify. You will obtain 3√a^3=a. That is what you do with the numerator.

However, multiplying the bottom of a fraction alone would affect the situation. You must multiply by the integer equivalent of 1:

1/3√a= 1/3√a * 3√a^2/ 3√a^2

=3√a^2/ 3√a^3

=3√a^2 / a.

Some frequently asked questions

How do you rationalize a two term denominator?

The steps for doing rationalization on denominators with two terms are listed below.

  • Multiply both the numerator and the denominator by the conjugate of the denominator.
  • Distribute the numerator and denominator or use the FOIL approach.
  • We may multiply numbers within a radical by numbers within a radical and numbers outside a radical by numbers outside a radical. Then, mix similar phrases.
  • Now combine the acquired similar words and simplify the radical terms.
  • If feasible, transform the fraction to a simpler form.
  • For the outcome of the preceding step, we may need to lower each integer outside the radical by the same number.
  • We cannot lower the fraction if we cannot reduce each number outside the radical by the same number. Let’s look at an example to assist you understand.

How do you rationalize a denominator?

So, in order to rationalize the denominator, we must remove all radicals from the denominator.

Step 1: Multiply the numerator and denominator by a radical that removes the radical from the denominator.

Step 2: Ensure that all radicals are simplified.

Step 3: If necessary, simplify the fraction.

What is the rationalized form?

Rationalization is the process of eliminating imaginary numbers from an algebraic expression’s denominator. It is the process of transferring the radical (i.e., square root or cube root) from the fraction’s denominator to its denominator (numerator).

What is an example of rationalization?

For example, a person may justify a terrible mood or general nasty conduct by stating that heavy traffic hindered the morning trip. Someone who is passed up for a promotion may justify their sadness by arguing that they did not want so much responsibility after all.

Do you always need to rationalize the denominator?

No, technically. The general rationale for this is to have a standard form. When looking at trig ratios using radicals, for example, these are supplied with rationalized denominators, making it easy to spot these ratios when you rationalize the denominator in your calculations.

How do you rationalize three terms in the denominator?

We may use the same procedures we used to rationalize the denominator with two terms, but with a slight difference. Consider a denominator with the following three terms: a + b + c. By multiplying with its conjugate, a – b, we rationalize a denominator with two terms: a + b. We may use the same logic to rationalize a denominator with three terms by combining them as a + b + c = (a + b) + c. The difference of squares formula gives us: [(a + b) + c] × [(a + b) – c] = (a + b)2 − c2.

What is the rationalization of the denominator?

When a given fraction has a radical word or a surd in the denominator, we rationalize the denominator. Square root and cube root are two examples of radical words. If the denominator of a two-term mathematical statement contains a radical, we must multiply both the numerator and the denominator by the denominator’s conjugate. This is known as rationalization. In other words, rationalizing the denominator involves transferring the radical term (square root or cube root) to the numerator, resulting in a denominator that is a whole integer.

Why does the butterfly method not work?

Because they couldn’t utilize the butterfly approach to add three or four fractions at once, they added two fractions at a time. Instead of finding a common denominator for all four fractions, they discovered a new common denominator with each addition (and those denominators were not the LCDs).

How do you rationalize a denominator on a calculator?

The rationalized denominator calculator is a free online application that calculates the rationalized denominator for a given input. Any online rationalization of the denominator calculator tool speeds up and simplifies the calculations, displaying the answer in a fraction of a second. The following is the process for rationalizing the denominator calculator:

  • In the input area, enter the numerator and denominator values.
  • To obtain the output, click the “Rationalize Denominator” button.
  • The output field will reveal the result.

How do you rationalize a denominator example?

Justify the denominator of 2/√8-√7

Solution:

The following expression is: 2/√8-√7.

Because the conjugate of √a – √b is √a + √b, the conjugate of √8 – √7 is √8 + √7. When we multiply the numerator and denominator by √8 + √7, we obtain;

2/√8-√7 * √8+√7 / √8+√7

Using the FOIL approach and appropriate IDs,

2/√8+√7 / (√8)^2+(√7)^2

In terms of simplicity,

=2/√8+√7 / (8-7)

=2/√8+√7 / 1

=2/√8+√7

As a result, the denominator found here is reasonable.

You May Also Like

About the Author: mike