Download ArticleDownload ArticleThis article was reviewed by Joseph Meyer and by How.com.vn staff writer, Hannah Madden. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources.
This article has been viewed 119,627 times.
Traditionally, a radical or irrational number cannot be left in the denominator (the bottom) of a fraction. When a radical does appear in the denominator, you need to multiply the fraction by a term or set of terms that can remove that radical expression. While the use of calculators make rationalizing fractions a bit dated, this technique may still be tested in class.
Steps
Examine the fraction. A fraction is written correctly when there is no radical in the denominator. If the denominator contains a square root or other radical, you must multiply both the top and bottom by a number that can get rid of that radical. Note that the numerator can contain a radical. Don't worry about the numerator.[1]- We can see that there is a in the denominator.
Multiply the numerator and denominator by the radical in the denominator. A fraction with a monomial term in the denominator is the easiest to rationalize. Both the top and bottom of the fraction must be multiplied by the same term, because what you are really doing is multiplying by 1.Advertisement
Simplify as needed. The fraction has now been rationalized.[2]
- 1Examine the fraction. If your fraction contains a sum of two terms in the denominator, at least one of which is irrational, then you cannot multiply the fraction by it in the numerator and denominator.[3]
- To see why this is the case, write an arbitrary fraction where and are irrational. Then the expression contains a cross-term If at least one of and is irrational, then the cross-term will contain a radical.
- Let's see how this works with our example.
- As you can see, there's no way we can get rid of the in the denominator after doing this.
Multiply the fraction by the conjugate of the denominator. The conjugate of an expression is the same expression with the sign reversed.[4] For example, the conjugate of is- Why does the conjugate work? Going back to our arbitrary fraction multiplying by the conjugate in the numerator and denominator results in the denominator being The key here is that there are no cross-terms. Since both of these terms are being squared, any square roots will be eliminated.
Simplify as needed.[5]
Examine the problem. If you are asked to write the reciprocal of a set of terms containing a radical, you will need to rationalize before simplifying. Use the method for monomial or binomial denominators, depending on whichever applies to the problem.[6]
Write the reciprocal as it would usually appear. A reciprocal is created when you invert the fraction.[7] Our expression is actually a fraction. It's just being divided by 1.
Multiply by something that can get rid of the radical on the bottom. Remember, you're actually multiplying by 1, so you have to multiply both the numerator and denominator. Our example is a binomial, so multiply the top and bottom by the conjugate.[8]
Simplify as needed.- Do not be thrown off by the fact that the reciprocal is the conjugate. This is just a coincidence.
Examine the fraction. You can also expect to face cube roots in the denominator at some point, though they are rarer. This method also generalizes to roots of any index.[9]
Rewrite the denominator in terms of exponents. Finding an expression that will rationalize the denominator here will be a bit different because we cannot simply multiply by the radical.[10]
Multiply the top and bottom by something that makes the exponent in the denominator 1. In our case, we are dealing with a cube root, so multiply by Remember that exponents turn a multiplication problem into an addition problem by the property [11]- This can generalize to nth roots in the denominator. If we have we multiply the top and bottom by This will make the exponent in the denominator 1.
Simplify as needed.[12]- If you need to write it in radical form, factor out the
![How.com.vn English: {\frac {3}{{\sqrt[ {3}]{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a37ab9394065e1be3dca1eeb4e96021d86c08a2)
![How.com.vn English: 3^{{2/3}}=(3^{{2}})^{{1/3}}={\sqrt[ {3}]{9}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e059be31ff9a14ad50f7f395de6079811267a674)
Community Q&A
- QuestionHow do I rationalize with three terms?
Community AnswerSomething like 1/(1+root2 + root3)? If so, group as 1+(root2 + root3) and multiply through by the "difference of squares conjugate" 1-(root2 + root3). That makes the denominator -4 - root6, which is still irrational, but did improve from two irrational terms to only one. So repeat the same trick by multiplying through by -4+root6 and the denominator is rationalized. - QuestionIn your pictures, what does the point mean?
DonaganTop AnswererIf you're asking about the dots that are placed between various fractions, those are multiplication signs. For example, in the article's second image we see (7√3) / (2√7), then a dot, then (√7 / √7). That means we multiply the first fraction by the second fraction (numerator times numerator, and denominator times denominator), giving us (7√21) / 14, which simplifies to √21 / 2. (Incidentally, the article shows some other dots that are not between fractions. Those are merely "bullet points.") - QuestionHow can I rationalize the denominator with a cube root that has a variable?
Community AnswerIf it is a binomial expression, follow the steps outlined in method 2.
Tips
References
- ↑ https://www.youtube.com/watch?v=GS7q-fUDxmQ
- ↑ https://www.youtube.com/watch?v=GS7q-fUDxmQ
- ↑ https://www.chilimath.com/lessons/intermediate-algebra/rationalizing-the-denominator/
- ↑ https://www.chilimath.com/lessons/intermediate-algebra/rationalizing-the-denominator/
- ↑ https://www.chilimath.com/lessons/intermediate-algebra/rationalizing-the-denominator/
- ↑ https://www.khanacademy.org/math/algebra-home/alg-exp-and-log/miscellaneous-radicals/v/how-to-rationalize-a-denominator
- ↑ https://www.mathsisfun.com/reciprocal.html
- ↑ https://www.khanacademy.org/math/algebra-home/alg-exp-and-log/miscellaneous-radicals/v/how-to-rationalize-a-denominator
- ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut41_rationalize.htm
About This Article
To rationalize a denominator, start by multiplying the numerator and denominator by the radical in the denominator. Then, simplify the fraction if necessary. If you're working with a fraction that has a binomial denominator, or two terms in the denominator, multiply the numerator and denominator by the conjugate of the denominator. To get the conjugate, just reverse the sign in the expression. Then, simplify your answer as needed. To learn how to rationalize a denominator with a cube root, scroll down!
Did this article help you?
⚠️ Disclaimer:
Content from Wiki How English language website. Text is available under the Creative Commons Attribution-Share Alike License; additional terms may apply.
Wiki How does not encourage the violation of any laws, and cannot be responsible for any violations of such laws, should you link to this domain, or use, reproduce, or republish the information contained herein.
- - A few of these subjects are frequently censored by educational, governmental, corporate, parental and other filtering schemes.
- - Some articles may contain names, images, artworks or descriptions of events that some cultures restrict access to
- - Please note: Wiki How does not give you opinion about the law, or advice about medical. If you need specific advice (for example, medical, legal, financial or risk management), please seek a professional who is licensed or knowledgeable in that area.
- - Readers should not judge the importance of topics based on their coverage on Wiki How, nor think a topic is important just because it is the subject of a Wiki article.
