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Mathematics LibreTexts

8.4: Multiplying and Dividing Radical Expressions

  • Anonymous
  • LibreTexts

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Learning Objectives
  • Multiply radical expressions.
  • Divide radical expressions.
  • Rationalize the denominator.

Multiplying Radical Expressions

When multiplying radical expressions with the same index, we use the product rule for radicals. If a and b represent positive real numbers,

nanb=nab

Example 8.4.1

Multiply:

26

Solution:

This problem is a product of two square roots. Apply the product rule for radicals and then simplify.

26=26Multiplytheradicands.=12Simplify.=223=23

Answer:

23

Example 8.4.2

Multiply:

3936

Solution:

This problem is a product of cube roots. Apply the product rule for radicals and then simplify.

3936=396Multiplytheradiands.=354Simplify.=3332=332

Answer:

332

Often there will be coefficients in front of the radicals.

Example 8.4.3

Multiply:

2352

Solution:

Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands as follows.

2352=2532Multiplicationiscommutative.=106Multiplythecoefficientsandtheradicands.=106

Typically, the first step involving the application of the commutative property is not shown.

Answer:

106

Example 8.4.4

Multiply:

Solution:

Answer:

30x

Use the distributive property when multiplying rational expressions with more than one term.

Example 8.4.5

Multiply:

Solution:

Apply the distributive property and multiply each term by 43.

Answer:

Example 8.4.6

Multiply:

Solution:

Apply the distributive property and then simplify the result.

Answer:

The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. Apply the distributive property, simplify each radical, and then combine like terms.

Example 8.4.7

Multiply:

(5+2)(54)

Solution:

Begin by applying the distributive property.

Screenshot (306).png
Figure 8.4.1

Answer:

325

Example 8.4.8

Multiply:

(3xy)2

Solution:

Answer:

9x6xy+y

Exercise 8.4.1

Multiply:

(23+52)(326)

Answer

6122+56203

The expressions (a+b) and (ab) are called conjugates. When multiplying conjugates, the sum of the products of the inner and outer terms results in 0.

Example 8.4.9

Multiply:

(2+5)(25)

Solution:

Apply the distributive property and then combine like terms.

Answer:

3

It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. This is true in general and is often used in our study of algebra.

(a+b)(ab)=a2ab+abb2=ab

Therefore, for nonnegative real numbers a and b, we have the following property:

(a+b)(ab)=ab

Dividing Radical Expressions (Rationalizing the Denominator)

To divide radical expressions with the same index, we use the quotient rule for radicals. If a and b represent nonnegative numbers, where b0, then we have

nanb=nab

Example 8.4.10

Divide:

8010

Solution:

In this case, we can see that 10 and 80 have common factors. If we apply the quotient rule for radicals and write it as a single square root, we will be able to reduce the fractional radicand.

8010=8010Applythequotientruleforradicalsandreducetheradicand.=8Simplify.=42=22

Answer:

22

Example 8.4.11

Divide:

16x5y42xy

Solution:

16x5y42xy=16x5y42xyApplythequotientruleforradicalsandcancel.=8x4y3Simplify.=42(x2)2y2y=2x2y2y

Answer:

2x2y2y

Example 8.4.12

Divide:

354a3b5316a2b2

Solution:

354a3b5316a2b2=354a3b516a2b2Applythequotientruleforradicalsandthencancel.=327ab38Replace27and8withtheirprimefactorizations.=333ab323Simplify.=333ab3323=3b3a2

Answer:

3b3a2

When the divisor of a radical expression contains a radical, it is a common practice to find an equivalent expression where the denominator is a rational number. Finding such an equivalent expression is called rationalizing the denominator.

RadicalexpressionRationaldenominator13=33

To do this, multiply the fraction by a special form of 1 so that the radicand in the denominator can be written with a power that matches the index. After doing this, simplify and eliminate the radical in the denominator. For example,

13=1333=332=33

Remember, to obtain an equivalent expression, you must multiply the numerator and denominator by the exact same nonzero factor.

Example 8.4.13

Rationalize the denominator:

32

Solution:

The goal is to find an equivalent expression without a radical in the denominator. In this example, multiply by 1 in the form 22.

32=3222Multiplyby22.=622Simplify.=62Rationaldenominator.

Answer:

62

Example 8.4.14

Rationalize the denominator:

123x

Solution:

The radicand in the denominator determines the factors that you need to use to rationalize it. In this example, multiply by 1 in the form 123x.

123x=123x3x3xMultiplyby3x3x.=3x232x2Simplify.=3x23x=3x6x

Answer:

3x6x

Typically, we will find the need to reduce, or cancel, after rationalizing the denominator.

Example 8.4.15

Rationalize the denominator:

525ab

Solution

In this example, we will multiply by 1 in the form 5ab5ab.

525ab=525ab5ab5ab=510ab25a2b2Simplify.=510ab5abCancel.=10abab

Notice that a and b do not cancel in this example. Do not cancel factors inside a radical with those that are outside.

Answer:

10abab

Exercise 8.4.2

Rationalize the denominator:

4a3b

Answer

23ab3b

Up to this point, we have seen that multiplying a numerator and a denominator by a square root with the exact same radicand results in a rational denominator. In general, this is true only when the denominator contains a square root. However, this is not the case for a cube root. For example,

13x3x3x=3x3x2

Note that multiplying by the same factor in the denominator does not rationalize it. In this case, if we multiply by 1 in the form of 3x23x2, then we can write the radicand in the denominator as a power of 3. Simplifying the result then yields a rationalized denominator. For example,

13x=13x3x23x2=3x23x3=3x2x

Therefore, to rationalize the denominator of radical expressions with one radical term in the denominator, begin by factoring the radicand of the denominator. The factors of this radicand and the index determine what we should multiply by. Multiply numerator and denominator by the n th root of factors that produce n th powers of all the factors in the radicand of the denominator.

Example 8.4.16

Rationalize the denominator:

1325

Solution:

The radical in the denominator is equivalent to 352. To rationalize the denominator, it should be 353. To obtain this, we need one more factor of 5. Therefore, multiply by 1 in the form of 3535.

1325=13523535Multiplybythecuberootoffactorsthatresultinpowersof3.=35353Simplify.=355

Answer:

355

Example 8.4.17

Rationalize the denominator:

327a2b2

Solution:

In this example, we will multiply by 1 in the form 322b322b.

327a2b2=333a32b2Applythequotientruleforradicals.=33a32b2322b322bMultiplybythecuberootoffactorsthatresultinpowersof3.=3322ab323b3Simplify.=334ab2b

Answer:

334ab2b

Example 8.4.18

Rationalize the denominator:

154x3

Solution:

In this example, we will multiply by 1 in the form 523x2523x2.

154x3=1522x3523x2523x2Multiplybythefifthrootoffactorsthatresultinpowersof5.=523x5525x5Simplify.=58x22x

Answer:

58x22x

When two terms involving square roots appear in the denominator, we can rationalize it using a very special technique. This technique involves multiplying the numerator and the denominator of the fraction by the conjugate of the denominator. Recall that multiplying a radical expression by its conjugate produces a rational number.

Example 8.4.19

Rationalize the denominator:

132

Solution:

In this example, the conjugate of the denominator is 3+2. Therefore, multiply by 1 in the form (3+2)(3+2).

132=1(32)(3+2)(3+2)Multiplynumeratoranddenominatorbytheconjugateofthedenominator.=3+29+664Simplify.=3+232=3+21=3+2

Answer:

3+2

Notice that the terms involving the square root in the denominator are eliminated by multiplying by the conjugate. We can use the property (a+b)(ab)=ab to expedite the process of multiplying the expressions in the denominator.

Example 8.4.20

Rationalize the denominator:

262+6

Solution:

Multiply by 1 in the form (26)(26)

Answer:

2+3

Example 8.4.21

Rationalize the denominator:

xyx+y

Solution:

In this example, we will multiply by 1 in the form (xy)(xy).

Answer:

Exercise 8.4.3

Rationalize the denominator:

35+5253

Answer

45+19511

Key Takeaways

  • To multiply two single-term radical expressions, multiply the coefficients and multiply the radicands. If possible, simplify the result.
  • Apply the distributive property when multiplying radical expressions with multiple terms. Then simplify and combine all like radicals.
  • Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression.
  • It is common practice to write radical expressions without radicals in the denominator. The process of finding such an equivalent expression is called rationalizing the denominator.
  • If an expression has one term in the denominator involving a radical, then rationalize it by multiplying numerator and denominator by the nth root of factors of the radicand so that their powers equal the index.
  • If a radical expression has two terms in the denominator involving square roots, then rationalize it by multiplying the numerator and denominator by its conjugate.
Exercise 8.4.4 multiplying radical expressions

Multiply. (Assume all variables are nonnegative.)

  1. 35
  2. 73
  3. 26
  4. 515
  5. 77
  6. 1212
  7. 25710
  8. 31526
  9. (25)2
  10. (62)2
  11. 2x2x
  12. 5y5y
  13. 3a12
  14. 3a2a
  15. 42x36x
  16. 510y22y
  17. 35325
  18. 3432
  19. 34310
  20. 31836
  21. (539)(236)
  22. (234)(334)
  23. (232)3
  24. (334)3
  25. 33a239a
  26. 37b349b2
  27. 36x234x2
  28. 312y39y2
  29. 320x2y310x2y2
  30. 363xy312x4y2
  31. 5(35)
  32. 2(32)
  33. 37(273)
  34. 25(6310)
  35. 6(32)
  36. 15(5+3)
  37. x(x+xy)
  38. y(xy+y)
  39. 2ab(14a210b)
  40. 6ab(52a3b)
  41. (25)(3+7)
  42. (3+2)(57)
  43. (234)(36+1)
  44. (526)(723)
  45. (53)2
  46. (72)2
  47. (23+2)(232)
  48. (2+37)(237)
  49. (a2b)2
  50. (ab+1)2
  51. What are the perimeter and area of a rectangle with length of 53 centimeters and width of 32 centimeters?
  52. What are the perimeter and area of a rectangle with length of 26 centimeters and width of 3 centimeters?
  53. If the base of a triangle measures 62 meters and the height measures 32 meters, then what is the area?
  54. If the base of a triangle measures 63 meters and the height measures 36 meters, then what is the area?
Answer

1. 15

3. 23

5. 7

7. 702

9. 20

11. 2x

13. 6a

15. 24x3

17. 5

19. 235

21. 3032

23. 16

25. 3a

27. 233(x2)23

29. 2xy325x

31. 355

33. 42321

35. 3223

37. x+xy

39. 2(14a210b)ab

41. 6+141535

43. 182+231264

45. 8215

47. 10

49. (a2b)2

51. Perimeter: (103+62) centimeters; area: 156 square centimeters

53. 18 square meters

Exercise 8.4.5 dividing radical expressions

Divide.

  1. 753
  2. 36010
  3. 7275
  4. 9098
  5. 90x52x
  6. 96y33y
  7. 162x7y52xy
  8. 363x4y93xy
  9. 316a5b232a2b2
  10. 3192a2b732a2b2
Answer

1. 5

3. 265

5. 35x2

7. 9x3y2

9. 2a

Exercise 8.4.6 dividing radical expressions

Rationalize the denominator

  1. 15
  2. 16
  3. 23
  4. 37
  5. 5210
  6. 356
  7. 353
  8. 622
  9. 17x
  10. 13y
  11. a5ab
  12. \(\3 b^{2} \sqrt{\frac{23}{a b}})
  13. 236−−√3
  14. 147√3
  15. 14x−−√3
  16. 13y2−−−−√3
  17. 9x⋅2√39xy2−−−−−√3
  18. 5y2⋅x−−√35x2y−−−−−√3
  19. 3a2 3a2b2−−−−−√3
  20. 25n3 25m2n−−−−−−√3
  21. 327x2y−−−−−√5
  22. 216xy2−−−−−−√5
  23. ab9a3b−−−−√5
  24. abcab2c3−−−−−√5
  25. 310−−√−3
  26. 26√−2
  27. 15√+3√
  28. 17√−2√
  29. 3√3√+6√
  30. 5√5√+15−−√
  31. 105−35√
  32. −22√4−32√
  33. 3√+5√3√−5√
  34. 10−−√−2√10−−√+2√
  35. 23√−32√43√+2√
  36. 65√+225√−2√
  37. x+y√x−y√
  38. x−y√x+y√
  39. a√−b√a√+b√
  40. ab−−√+2√ab−−√−2√
  41. x−−√5−2x−−√ 106. 1x−−√−y
Answer

1. 55

3. 63

5. 1305

7. 3153

9. 7x7x

11. a5abab

13. 6√33

15. 2x2−−−−√32x

17. 3 6x2y−−−−−√3y

19. 9ab−−−√32b

21. 9x3y4−−−−−−√5xy

23. 27a2b4−−−−−−√53

25. 310−−√+9

27. 5√−3√2

29. −1+2√

31. −5−35√2

33. −4−15−−√

35. 15−76√23

37. x2+2xy√+yx2−y

39. a−2ab−−√+ba−b

41. 5x−−√+2x25−4x

Exercise 8.4.7 discussion
  1. Research and discuss some of the reasons why it is a common practice to rationalize the denominator.
  2. Explain in your own words how to rationalize the denominator.
Answer

1. Answer may vary


This page titled 8.4: Multiplying and Dividing Radical Expressions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.

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