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10.4: Rationalize Denominators

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Definition: Rationalizing the Denominator

Rationalizing the denominator is the process for obtaining denominators without radicals.

When given a quotient with radicals, it is common practice to leave an expression without a radical in the denominator. After simplifying an expression, if there is a radical in the denominator, we will rationalize it so that the denominator is left without any radicals. We start by rationalizing denominators with square roots, and then extend this idea to higher roots.

Rationalizing Denominators with Square Roots

Rationalizing the Denominator with Square Roots

To rationalize the denominator with a square root, multiply the numerator and denominator by the exact radical in the denominator, e.g, 1xxx

Example 10.4.1

Simplify: 65

Solution

We see the expression is irreducible and that the denominator contains 5. We rationalize the denominator so that the denominator is left without radicals.

65Rationalize the denominator6555Multiply fractions6555Apply product rule3025Simplify radicals305Simplified expression

Notice, the expression is simplified completely and there are no longer any radicals in the denominator. This is the goal for these problems.

Example 10.4.2

Simplify: 6141222

Solution

We see the expression isn’t reduced. We will reduce the fraction by applying the quotient rule, then rationalize the denominator, if needed.

6141222Apply quotient rule6121422Reduce fractions12711Rewrite as one fraction7211Rationalize the denominator72111111Multiple fractions71121111Apply product rule772121Simplify radicals77211Simplify radicals7722Simplified expression

Example 10.4.3

Simplify: 3926

Solution

We see the expression is irreducible and that the denominator contains 6. We rationalize the denominator so that the denominator is left without radicals.

3926Rationalize the denominator(39)2666Multiply fractions6(39)266Distribute and apply product rule1896236Rewrite the radicand 18929612Simplify radicals329612Factor a GCF from the numerator3(236)124Reduce by a factor of 3(236)4Simplified expression

Rationalizing Denominators with Higher Roots

Radicals with higher roots in the denominators are a bit more challenging. Notice, rationalizing the denominator with square roots works out nicely because we are only trying to obtain a radicand that is a perfect square in the denominator. Here, we are trying to obtain radicands that are perfect cubes or higher in the denominator. Let’s try an example.

Example 10.4.4

Simplify: 4327325

Solution

We see the expression is irreducible and that the denominator contains 325. We rationalize the denominator so that the denominator is left without radicals. Notice we need a radicand that is a perfect cube in the denominator.

4327325Rationalize the denominator43273253535Multiply fractions43235732535Apply product rule431073125Simplify radicals431075Simplify radicals431035Simplified expression

We choose to multiply by 35 because we noticed 325=352, and all we needed was an additional factor of 5 to make a perfect cube in the denominator. Since 7 is a coefficient and not a part of the radicand, we do not include it when rationalizin

Example 10.4.5

Simplify: 341142

Solution

We see the expression is irreducible and that the denominator contains 42. We rationalize the denominator so that the denominator is left without radicals. Notice we need a radicand that is a perfect fourth power in the denominator.

341142Rationalize the denominator3411424848Multiply fractions3411484248Apply product rule3488416Simplify radicals34882Simplified expression

We choose to multiply by 48 because we noticed 42, and all we needed was three additional factors of 2 to make a perfect fourth power in the denominator.

Rationalize Denominators Using the Conjugate

There are times where the given denominator is not just one term. Often, in the denominator, we have a difference or sum of two terms in which one or both terms are square roots. In order to rationalize these denominators, we use the idea from a difference of two squares:

(a+b)(ab)=a2b2

Notice, with the difference of two squares, we are left without any outer or inner product terms- just the squares of the first and last terms. Since these denominators take the form of a binomial, we have a special name for the factor we use when rationalizing the denominator. The factor is called the conjugate.

Rationalize Denominators Using the Conjugate

We rationalize denominators of the type a±b by multiplying the numerator and denominator by their conjugates, e.g.,

1a+babab

The conjugate for

  • a+b is ab
  • ab is a+b

The case is similar for when there is something like a±b in the denominator.

Putting all these ideas together, let's try an example.

Example 10.4.6

Simplify: 235

Solution

We notice the difference in the denominator and so we know we will use the conjugate to rationalize the denominator.

235Rationalize the denominator2(35)(3+5)(3+5)Multiply fractions2(3+5)(35)(3+5)Distribute and FOIL23+109+535325Simplify23+10325Subtract23+1022Factor a GCF from the numerator2(3+5)22Reduce by a factor of 23+511Rewrite3+511Simplified expression

Example 10.4.7

Simplify: 3523

Solution

We notice the difference in the denominator and so we know we will use the conjugate to rationalize the denominator.

3523Rationalize the denominator(35)(23)(2+3)(2+3)Multiply fractions(35)(2+3)(23)(2+3)FOIL6+3325154+23239Simplify6+33251543Subtract6+3325151Rewrite6+332515Simplified expression

Note

During the 5th century BC in India, Aryabhata published a treatise on astronomy. His work included a method for finding the square root of numbers that have many digits.

Example 10.4.8

Simplify: 253756+42

Solution

We notice the sum in the denominator and so we know we will use the conjugate to rationalize the denominator.

253756+42Rationalize the denominator(2537)(56+42)(5642)(5642)Multiply fractions(2537)(5642)(56+42)(5642)FOIL10308101542121425362012+2012164Simplify103081015421214256162Subtract103081015421214118Simplified expression

Rationalize Denominators Homework

Simplify.

Exercise 10.4.1

2433

Exercise 10.4.2

35434

Exercise 10.4.3

123

Exercise 10.4.4

235

Exercise 10.4.5

4315

Exercise 10.4.6

4+239

Exercise 10.4.7

4+2354

Exercise 10.4.8

255413

Exercise 10.4.9

2333

Exercise 10.4.10

535+2

Exercise 10.4.11

25+2

Exercise 10.4.12

3433

Exercise 10.4.13

43+5

Exercise 10.4.14

4442

Exercise 10.4.15

11+2

Exercise 10.4.16

14272

Exercise 10.4.17

ababa

Exercise 10.4.18

a+aba+b

Exercise 10.4.19

2+62+3

Exercise 10.4.20

aba+b

Exercise 10.4.21

63223

Exercise 10.4.22

ababba

Exercise 10.4.23

253+5

Exercise 10.4.24

4+349

Exercise 10.4.25

232216

Exercise 10.4.26

5+4417

Exercise 10.4.27

5236

Exercise 10.4.28

53+45

Exercise 10.4.29

5232

Exercise 10.4.30

422

Exercise 10.4.31

225+23

Exercise 10.4.32

4435

Exercise 10.4.33

3+331

Exercise 10.4.34

2+102+5

Exercise 10.4.35

14714+7

Exercise 10.4.36

a+aba+b

Exercise 10.4.37

25+313

Exercise 10.4.38

aba+b

Exercise 10.4.39

ababba

Exercise 10.4.40

42+332+3

Exercise 10.4.41

1+525+52

Exercise 10.4.42

52+35+52

Exercise 10.4.43

3+2232


This page titled 10.4: Rationalize Denominators is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

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