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8.6: Divide Radical Expressions

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Learning Objectives

By the end of this section, you will be able to:

  • Divide radical expressions
  • Rationalize a one term denominator
  • Rationalize a two term denominator

Before you get started, take this readiness quiz.

  1. Simplify: 3048.
    If you missed this problem, review Example 1.24.
  2. Simplify: x2x4.
    If you missed this problem, review Example 5.12.
  3. Multiply: (7+3x)(73x).
    If you missed this problem, review Example 5.32.

Divide Radical Expressions

We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals. We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars re needed.

Definition 8.6.1: Quotient Property of Radical Expressions

If na and nb are real numbers, b0, and for any integer n2 then,

nab=nanb and nanb=nab

We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.

Example 8.6.1

Simplify:

  1. 72x3162x
  2. 332x234x5

Solution:

a.

72x3162x

Rewrite using the quotient property,

72x3162x

Remove common factors.

184x2x189x

Simplify.

4x29

Simplify the radical.

2x3

b.

332x234x5

Rewrite using the quotient property, nanb=nab.

332x24x5

Simplify the fraction under the radical.

38x3

Simplify the radical.

2x

Exercise 8.6.1

Simplify:

  1. 50s3128s
  2. 356a37a4
Answer
  1. 5s8
  2. 2a
Exercise 8.6.2

Simplify:

  1. 75q5108q
  2. 372b239b5
Answer
  1. 5q26
  2. 2b
Example 8.6.2

Simplify:

  1. 147ab83a3b4
  2. 3250mn232m2n4

Solution:

a.

147ab83a3b4

Rewrite using the quotient property.

147ab83a3b4

Remove common factors in the fraction.

49b4a2

Simplify the radical.

7b2a

b.

3250mn232m2n4

Rewrite using the quotient property.

3250mn22m2n4

Simplify the fraction under the radical.

3125m3n6

Simplify the radical.

5mn2

Exercise 8.6.3

Simplify:

  1. 162x10y22x6y6
  2. 3128x2y132x1y2
Answer
  1. 9x2y2
  2. 4xy
Exercise 8.6.4

Simplify:

  1. 300m3n73m5n
  2. 381pq133p2q5
Answer
  1. 10n3m
  2. 3pq2
Example 8.6.3

Simplify: 54x5y33x2y

Solution:

54x5y33x2y

Rewrite using the quotient property.

54x5y33x2y

Remove common factors in the fraction.

18x3y2

Rewrite the radicand as a product using the largest perfect square factor.

9x2y22x

Rewrite the radical as the product of two radicals.

9x2y22x

Simplify.

3xy2x

Exercise 8.6.5

Simplify: 64x4y52xy3

Answer

4xy2x

Exercise 8.6.6

Simplify: 96a5b42a3b

Answer

4ab3b

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process!

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator. This process is still used today, and is useful in other areas of mathematics, too.

Definition 8.6.2: Rationalizing the Denominator

Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a radical.

Similarly, a radical expression is not considered simplified if the radicand contains a fraction.

Simplified Radical Expressions

A radical expression is considered simplified if there are

  • no factors in the radicand have perfect powers of the index
  • no fractions in the radicand
  • no radicals in the denominator of a fraction

To rationalize a denominator with a square root, we use the property that (a)2=a. If we square an irrational square root, we get a rational number.

We will use this property to rationalize the denominator in the next example.

Example 8.6.4

Simplify:

  1. 43
  2. 320
  3. 36x

Solution:

To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

a.

 

.

Multiply both the numerator and denominator by 3.

.

Simplify.

.

Table 8.5.1

b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.

 

.

The fraction is not a perfect square, so rewrite using the Quotient Property.

.

Simplify the denominator.

.

Multiply the numerator and denominator by 5.

.

Simplify.

.

Simplify.

.

Table 8.5.2

c.

 

.

Multiply the numerator and denominator by 6x.

.

Simplify.

.

Simplify.

.

Table 8.5.3
Exercise 8.6.7

Simplify:

  1. 53
  2. 332
  3. 22x
Answer
  1. 533
  2. 68
  3. 2xx
Exercise 8.6.8

Simplify:

  1. 65
  2. 718
  3. 55x
Answer
  1. 655
  2. 146
  3. 5xx

When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.

We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.

For example,

Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.
Figure 8.5.14

We will use this technique in the next examples.

Example 8.6.5

Simplify:

  1. 136
  2. 3724
  3. 334x

Solution:

To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

a.

 

.

The radical in the denominator has one factor of 6. Multiply both the numerator and denominator by 362, which gives us 2 more factors of 6.

.

Multiply. Notice the radicand in the denominator has 3 powers of 6.

.

Simplify the cube root in the denominator.

.

Table 8.5.4

b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.

 

.

The fraction is not a perfect cube, so rewrite using the Quotient Property.

.

Simplify the denominator.

.

Multiply the numerator and denominator by 332. This will give us 3 factors of 3.

.

Simplify.

.

Remember, 333=3.

.

Simplify.

.

Table 8.5.5

c.

 

.

Rewrite the radicand to show the factors.

.

Multiple the numerator and denominator by 32x2. This will get us 3 factors of 2 and 3 factors of x.

.

Simplify.

.

Simplify the radical in the denominator.

.

Table 8.5.6
Exercise 8.6.9

Simplify:

  1. 137
  2. 3512
  3. 539y
Answer
  1. 3497
  2. 3906
  3. 533y23y
Exercise 8.6.10

Simplify:

  1. 132
  2. 3320
  3. 2325n
Answer
  1. 342
  2. 315010
  3. 235n25n
Example 8.6.6

Simplify:

  1. 142
  2. 4564
  3. 248x

Solution:

To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

a.

 

.

The radical in the denominator has one factor of 2.
Multiply both the numerator and denominator by 423, which gives us 3 more factors of 2.

.

Multiply. Notice the radicand in the denominator has 4 powers of 2.

.

Simplify the fourth root in the denominator.

.

Table 8.5.7

b. We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.

 

.

The fraction is not a perfect fourth power, so rewrite using the Quotient Property.

.

Rewrite the radicand in the denominator to show the factors.

.

Simplify the denominator.

.

Multiply the numerator and denominator by 422. This will give us 4 factors of 2.

.

Simplify.

.

Remember, 424=2.

.

Simplify.

.

Table 8.5.8

c.

 

.

Rewrite the radicand to show the factors.

.

Multiply the numerator and denominator by 42x3. This will get us 4 factors of 2 and 4 factors of x.

.

Simplify.

.

Simplify the radical in the denominator.

.

Simplify the fraction.

.

Table 8.5.9
Exercise 8.6.11

Simplify:

  1. 143
  2. 4364
  3. 34125x
Answer
  1. 4273
  2. 4124
  3. 345x35x
Exercise 8.6.12

Simplify:

  1. 145
  2. 47128
  3. 444x
Answer
  1. 41255
  2. 42248
  3. 464x3x

Rationalize a Two Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates Pattern to rationalize the denominator.

(ab)(a+b)(25)(2+5)a2b222(5)2451

When we multiple a binomial that includes a square root by its conjugate, the product has no square roots.

Example 8.6.7

Simplify: 523

Solution:

  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
Simplify the denominator. .
Simplify. .
Table 8.5.10
Exercise 8.6.13

Simplify: 315.

Answer

3(1+5)4

Exercise 8.6.14

Simplify: 246.

Answer

4+65

Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.

Example 8.6.8

Simplify: 3u6.

Solution:

  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
Table 8.5.11
Exercise 8.6.15

Simplify: 5x+2.

Answer

5(x2)x2

Exercise 8.6.16

Simplify: 10y3

Answer

10(y+3)y3

Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.

Example 8.6.9

Simplify: x+7x7.

Solution:

  .
Multiply the numerator and denominator by the conjugate of the denominator. .
Multiply the conjugates in the denominator. .
Simplify the denominator. .
Table 8.5.12

We do not square the numerator. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.

Exercise 8.6.17

Simplify: p+2p2.

Answer

(p+2)2p2

Exercise 8.6.18

Simplify: q10q+10

Answer

(q10)2q10

Key Concepts

  • Quotient Property of Radical Expressions
    • If na and nb are real numbers, b0, and for any integer n2 then, nab=nanb and nanb=nab
  • Simplified Radical Expressions
    • A radical expression is considered simplified if there are:
      • no factors in the radicand that have perfect powers of the index
      • no fractions in the radicand
      • no radicals in the denominator of a fraction

Glossary

rationalizing the denominator
Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.

This page titled 8.6: Divide Radical Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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