Menu

Math Lesson 7.5.2 - Rationalising the Denominator

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

[ 1 Votes ]

Welcome to our Math lesson on Rationalising the Denominator, this is the second lesson of our suite of math lessons covering the topic of Rationalising the Denominator, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Rationalising the Denominator

Now, let's come to the point of this tutorial. In algebra, it is not suitable to have a root in the denominator of a fraction. (We have experienced a similar situation when writing numbers in standard form, where it is not suitable to express the part leading the powers of ten by a number less than 1 or more than 10). Therefore, if the denominator of a fraction contains a radical (root) that belongs to the subset of surds, we must find the way to get rid of it. We call this action 'rationalising the denominator'.

It is worth to note here that we are interested in eliminating only the root from the denominator. As for the numerator, it is not a problem if it contains a root.

There are three methods for rationalising the denominator of a fraction containing surds. All of them are explained below illustrated with examples.

1. The denominator contains a single term with a root

When denominator is made of a single term containing a root, we simply multiply up and down (numerator and denominator) by the root itself as many times as needed, depending on the root index. This eliminates the root from denominator. Recall from properties of fractions that we can multiply the terms of a fraction by the same number to obtain an equivalent fraction.

The formula used in this case for rationalising the denominator is

a/b√x = a√x/b√x ∙ √x = a√x/bx

For example,

5/2√3 = 5√3/2√3 ∙ √3
= 5√3/2 ∙ 3
= 5√3/6

Example 2

Rationalise the denominator of the following fractions.

  1. 2/√5
  2. 2√3/√6

Solution 2

  1. Multiplying up and down by √5 yields
    2/√5 = 2√5/√5 ∙ √5
    = 2√5/5
  2. Multiplying up and down by √6 yields
    (2√3)/√6 = (2√3 ∙ √6)/(√6 ∙ √6)
    = 2√(3 ∙ 6)/(√6)2
    = 2√18/6
    Despite the fact that the denominator is already rationalised, we continue the operations until the fraction is expressed in the simplest terms. Thus, we write
    2√18/6 = 2√(9 ∙ 2)/6
    = 2√9 ∙ √2/6
    = 2 ∙ 3 ∙ √2/6
    = 6√2/6
    = √2
    We could have used a shorter method in this example. Thus,
    2√3/√6 = 2√3/√(2 ∙ 3)
    = 2√3/√2 ∙ √3
    = 2/√2
    = 2√2/√2 ∙ √2
    = 2√2/2
    = √2

The denominator contains a binomial where at least one of terms contains a surd

In this case, we multiply up and down by the conjugate of denominator. This allows get rid of the surd in the denominator as it raises at the second power. For example, if we consider the expression

2/3 - √5

the first thing to do is multiplying up and down by 3 + √5. This means making use of the third special algebraic identity to eliminate the root by raising it to the second power. Hence,

2/3 - √5
= 2 ∙ (3 + √5)/(3 - √5)(3 + √5)
= 2 ∙ (3 + √5)/32 - (√5)2
= 2 ∙ (3 + √5)/9 - 5
= 2 ∙ (3 + √5)/4
= 3 + √5/2

Example 3

Rationalise the denominator of the following fractions and if necessary continue the operations until you obtain an expression in the simplest terms.

  1. 6/√2 + √5
  2. 2 - √3/5√3 + 1

Solution

  1. Multiplying up and down by the conjugate of denominator yields
    6/√2 + √5
    = 6 ∙ (√2 - √5)/(√2 + √5)(√2 - √5)
    = 6 ∙ (√2 - √5)/(√2)2 -(√5)2
    = 6 ∙ (√2 - √5)/2 - 5
    = 6 ∙ (√2 - √5)/-3
    = -2 ∙ (√2 - √5)
  2. Multiplying up and down by the conjugate of denominator yields
    2 - √3/5√3 + 1
    = (2 - √3)(5√3 - 1)/(5√3 + 1)(5√3 - 1)
    = 2 ∙ 5√3 + 2 ∙ (-1) - √3 ∙ 5√3 - √3 ∙ (-1)/(5√3)2 -12
    = 10√3 - 2 - 5 ∙ (√3)2 + √3/52 ∙ (√3)2 - 12
    = 10√3 - 2-5 ∙ 3 + √3/25 ∙ 3 - 1
    = 11√3 - 17/74

3 Rationalising the denominator when an entire fraction is inside a root

When we have the root of an entire fraction involved, the first thing to do is to apply the rule

√(a/b) = √a/√b

where b can also be a binomial, not just a number. Then, we use the other two methods described earlier to rationalise the denominator. For example,

√(18/25) = √18/√25
= √9 ∙ √2/√25
= √3 ∙ √3 ∙ √2/√5 ∙ √5
= 3√2/5

Example 4

Rationalise the denominators of the following expressions.

  1. √(75/32)
  2. √(45/72)

Solution 4

  1. We have
    √(75/32) = √75/√32
    = √25 ∙ √3/√16 ∙ √2
    = 5√3/4√2
    = 5√3 ∙ √2/4√2 ∙ √2
    = 5√6/4 ∙ 2
    = 5√6/8
  2. We have
    √(45/72) = √45/√72
    = √9 ∙ √5/√36 ∙ √2
    = √(32 ) ∙ √5/√(62 ) ∙ √2
    = 3√5/6√2
    = √5/3√2
    = √5 ∙ √2/3√2 ∙ √2
    = √10/3 ∙ 2
    = √10/6

More Rationalising the Denominator Lessons and Learning Resources

Powers and Roots Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
7.5Rationalising the Denominator
Lesson IDMath Lesson TitleLessonVideo
Lesson
7.5.1Multiplying Brackets containing Surds
7.5.2Rationalising the Denominator

Whats next?

Enjoy the "Rationalising the Denominator" math lesson? People who liked the "Rationalising the Denominator lesson found the following resources useful:

  1. Rationalising Feedback. Helps other - Leave a rating for this rationalising (see below)
  2. Powers and Roots Math tutorial: Rationalising the Denominator. Read the Rationalising the Denominator math tutorial and build your math knowledge of Powers and Roots
  3. Powers and Roots Video tutorial: Rationalising the Denominator. Watch or listen to the Rationalising the Denominator video tutorial, a useful way to help you revise when travelling to and from school/college
  4. Powers and Roots Revision Notes: Rationalising the Denominator. Print the notes so you can revise the key points covered in the math tutorial for Rationalising the Denominator
  5. Powers and Roots Practice Questions: Rationalising the Denominator. Test and improve your knowledge of Rationalising the Denominator with example questins and answers
  6. Check your calculations for Powers and Roots questions with our excellent Powers and Roots calculators which contain full equations and calculations clearly displayed line by line. See the Powers and Roots Calculators by iCalculator™ below.
  7. Continuing learning powers and roots - read our next math tutorial: Indices

Help others Learning Math just like you

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

[ 1 Votes ]

We hope you found this Math tutorial "Rationalising the Denominator" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.

Powers and Roots Calculators by iCalculator™