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5.4 Multiplying and Dividing Radical Expressions

Learning Objectives

  1. Multiply radical expressions.
  2. Divide radical expressions.
  3. Rationalize the denominator.

Multiplying Radical Expressions

When multiplying radical expressions with the same index, we use the product rule for radicals. Given real numbers An and Bn,

AnBn=ABn

Example 1

Multiply: 12363.

Solution:

Apply the product rule for radicals, and then simplify.

12363=1263Multiplytheradicands.=723Simplify.=23323=2323=293

Answer: 293

Often, there will be coefficients in front of the radicals.

Example 2

Multiply: 3652

Solution:

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

3652=3562Multiplicationiscommutative.=1512Multiplythecoefficientsandtheradicands.=1543Simplify.=1523=303

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

Answer: 303

Example 3

Multiply: 34y23516y3.

Solution:

34y23516y3=1564y33Multiplythecoefficientsandthenmultiplytheradicands.=1543y33Simplify.=154y=60y

Answer: 60y

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

Example 4

Multiply: 52x(3x2x).

Solution:

Apply the distributive property and multiply each term by 52x.

52x(3x2x)=52x3x52x2xDistribute.=152x254x2Simplify.=15x252x=15x210x

Answer: 15x210x

Example 5

Multiply: 6x2y3(9x2y2354xy3).

Solution:

Apply the distributive property, and then simplify the result.

6x2y3(9x2y2354xy3)=6x2y39x2y236x2y354xy3=54x4y33524x3y23=272xx3y33583x3y23=3xy2x352x3y23=3xy2x310x3y23

Answer: 3xy2x310x3y23

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 6

Multiply: (x5y)2.

Solution:

(x5y)2=(x5y)(x5y)

Begin by applying the distributive property.

=xx+x(5y)+(5y)x+(5y)(5y)=x25xy5xy+25y2=x10xy+25y

Answer: x10xy+25y

The binomials (a+b) and (ab) are called conjugatesThe factors (a+b) and (ab) are conjugates.. When multiplying conjugate binomials the middle terms are opposites and their sum is zero.

Example 7

Multiply: (10+3)(103).

Solution:

Apply the distributive property, and then combine like terms.

(10+3)(103)=1010+10(3)+310+3(3)Distribute.=10030+309Simplify.=1030+303oppositesaddto0=103=7

Answer: 7

It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. This is true in general

(x+y)(xy)=x2xy+xyy2=xy

Alternatively, using the formula for the difference of squares we have,

(a+b)(ab)=a2b2Differenceofsquares.(x+y)(xy)=(x)2(y)2=xy

Try this! Multiply: (32y)(3+2y). (Assume y is positive.)

Answer: 94y

Dividing Radical Expressions

To divide radical expressions with the same index, we use the quotient rule for radicals. Given real numbers An and Bn,

AnBn=ABn

Example 8

Divide: 96363.

Solution:

In this case, we can see that 6 and 96 have common factors. If we apply the quotient rule for radicals and write it as a single cube root, we will be able to reduce the fractional radicand.

96363=9663Applythequotientruleforradicalsandreducetheradicand.=163Simplify.=823=223

Answer: 223

Example 9

Divide: 50x6y48x3y.

Solution:

Write as a single square root and cancel common factors before simplifying.

50x6y48x3y=50x6y48x3yApplythequotientruleforradicalsandcancel.=25x3y34Simplify.=25x3y34=5xyxy2

Answer: 5xyxy2

Rationalizing the Denominator

When the denominator (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 denominatorThe process of determining an equivalent radical expression with a rational denominator..

RadicalexpressionRationaldenominator12=22

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:

12=1222=24=22

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

Example 10

Rationalize the denominator: 25x.

Solution:

The goal is to find an equivalent expression without a radical in the denominator. 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 5x5x.

25x=25x5x5xMultiplyby5x5x.=10x25x2Simplify.=10x5x

Answer: 10x5x

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

Example 11

Rationalize the denominator: 3a26ab.

Solution:

In this example, we will multiply by 1 in the form 6ab6ab.

3a26ab=3a26ab6ab6ab=3a12ab36a2b2Simplify.=3a43ab6ab=6a3ab6abCancel.=3abb

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

Answer: 3abb

Try this! Rationalize the denominator: 9x2y.

Answer: 32xy2y

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, 1x3x3x3=x3x23 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 x23x23, then we can write the radicand in the denominator as a power of 3. Simplifying the result then yields a rationalized denominator.

1x3=1x3x23x23=x23x33=x23x

Therefore, to rationalize the denominator of a radical expression 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 the numerator and denominator by the nth root of factors that produce nth powers of all the factors in the radicand of the denominator.

Example 12

Rationalize the denominator: 23253.

Solution:

The radical in the denominator is equivalent to 523. To rationalize the denominator, we need: 533. To obtain this, we need one more factor of 5. Therefore, multiply by 1 in the form of 5353.

23253=235235353Multiplybythecuberootof  factorsthatresultinpowersof3.=103533Simplify.=1035

Answer: 1035

Example 13

Rationalize the denominator: 27a2b23.

Solution:

In this example, we will multiply by 1 in the form 22b322b3.

27a2b23=33a32b23Applythequotientruleforradicals.=3a32b2322b322b3Multiplybythecuberootoffactorsthatresultinpowersof3.=322ab323b33Simplify.=34ab32b

Answer: 34ab32b

Example 14

Rationalize the denominator: 2x554x3y5.

Solution:

In this example, we will multiply by 1 in the form 23x2y4523x2y45.

2x554x3y5=2x5522x3y523x2y4523x2y45Multiplybythefifthrootoffactorsthatresultinpowersof5.=2x523x2y4525x5y55Simplify.=2x40x2y452xy=40x2y45y

Answer: 40x2y45y

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 15

Rationalize the denominator: 153.

Solution:

In this example, the conjugate of the denominator is 5+3. Therefore, multiply by 1 in the form (5+3)(5+3).

153=1(53)(5+3)(5+3)Multiplynumeratoranddenominatorbytheconjugateofthedenominator.=5+325+15159Simplify.=5+353=5+32

Answer: 5+32

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 16

Rationalize the denominator: 102+6.

Solution:

Multiply by 1 in the form 2626.

102+6=(10)(2+6)(26)(26)Multiplybytheconjugateofthedenominator.=206026Simplify.=454154=252154=2(515)4=5152=5152=5+152

Answer: 1552

Example 17

Rationalize the denominator: xyx+y.

Solution:

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

xyx+y=(xy)(x+y)(xy)(xy)Multiplybytheconjugateofthedenominator.=x2xyxy+y2xySimplify.=x2xy+yxy

Answer: x2xy+yxy

Try this! Rationalize the denominator: 2353

Answer: 53+311

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 a radical expression 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 the 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 the conjugate of the denominator.

Topic Exercises

    Part A: Multiplying Radical Expressions

      Multiply. (Assume all variables represent non-negative real numbers.)

    1. 37

    2. 25

    3. 612

    4. 1015

    5. 26

    6. 515

    7. 77

    8. 1212

    9. 25710

    10. 31526

    11. (25)2

    12. (62)2

    13. 2x2x

    14. 5y5y

    15. 3a12

    16. 3a2a

    17. 42x36x

    18. 510y22y

    19. 3393

    20. 43163

    21. 153253

    22. 1003503

    23. 43103

    24. 18363

    25. (593)(263)

    26. (243)(343)

    27. (223)3

    28. (343)3

    29. 3a239a3

    30. 7b349b23

    31. 6x234x23

    32. 12y39y23

    33. 20x2y310x2y23

    34. 63xy312x4y23

    35. 5(35)

    36. 2(32)

    37. 37(273)

    38. 25(6310)

    39. 6(32)

    40. 15(5+3)

    41. x(x+xy)

    42. y(xy+y)

    43. 2ab(14a210b)

    44. 6ab(52a3b)

    45. 63(93203)

    46. 123(363+143)

    47. (25)(3+7)

    48. (3+2)(57)

    49. (234)(36+1)

    50. (526)(723)

    51. (53)2

    52. (72)2

    53. (23+2)(232)

    54. (2+37)(237)

    55. (a2b)2

    56. (ab+1)2

    57. What is the perimeter and area of a rectangle with length measuring 53 centimeters and width measuring 32 centimeters?

    58. What is the perimeter and area of a rectangle with length measuring 26 centimeters and width measuring 3 centimeters?

    59. If the base of a triangle measures 62 meters and the height measures 32 meters, then calculate the area.

    60. If the base of a triangle measures 63 meters and the height measures 36 meters, then calculate the area.

    Part B: Dividing Radical Expressions

      Divide. (Assume all variables represent positive real numbers.)

    1. 753
    2. 36010
    3. 7275
    4. 9098
    5. 90x52x
    6. 96y33y
    7. 162x7y52xy
    8. 363x4y93xy
    9. 16a5b232a2b23
    10. 192a2b732a2b23

    Part C: Rationalizing the Denominator

      Rationalize the denominator. (Assume all variables represent positive real numbers.)

    1. 15
    2. 16
    3. 23
    4. 37
    5. 5210
    6. 356
    7. 353
    8. 622
    9. 17x
    10. 13y
    11. a5ab
    12. 3b223ab
    13. 2363
    14. 1473
    15. 14x3
    16. 13y23
    17. 9x239xy23
    18. 5y2x35x2y3
    19. 3a23a2b23
    20. 25n325m2n3
    21. 327x2y5
    22. 216xy25
    23. ab9a3b5
    24. abcab2c35
    25. 3x8y2z5
    26. 4xy29x3yz45
    27. 3103
    28. 262
    29. 15+3
    30. 172
    31. 33+6
    32. 55+15
    33. 10535
    34. 22432
    35. 3+535
    36. 10210+2
    37. 233243+2
    38. 65+2252
    39. xyx+y
    40. xyxy
    41. x+yxy
    42. xyx+y
    43. aba+b
    44. ab+2ab2
    45. x52x
    46. 1xy
    47. x+2y2xy
    48. 3xyx+3y
    49. 2x+12x+11
    50. x+11x+1
    51. x+1+x1x+1x1
    52. 2x+32x32x+3+2x3
    53. The radius of the base of a right circular cone is given by r=3Vπh where V represents the volume of the cone and h represents its height. Find the radius of a right circular cone with volume 50 cubic centimeters and height 4 centimeters. Give the exact answer and the approximate answer rounded to the nearest hundredth.

    54. The radius of a sphere is given by r=3V4π3 where V represents the volume of the sphere. Find the radius of a sphere with volume 135 square centimeters. Give the exact answer and the approximate answer rounded to the nearest hundredth.

    Part D: 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.

Answers

  1. 21

  2. 62

  3. 23

  4. 7

  5. 702

  6. 20

  7. 2x

  8. 6a

  9. 24x3

  10. 3

  11. 533

  12. 253

  13. 3023

  14. 16

  15. 3a

  16. 2x3x3

  17. 2xy25x3

  18. 355

  19. 42321

  20. 3223

  21. x+xy

  22. 2a7b4b5a

  23. 3232153

  24. 6+141535

  25. 182+231264

  26. 8215

  27. 10

  28. a22ab+2b

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

  30. 18 square meters

  1. 5

  2. 265
  3. 3x25

  4. 9x3y2

  5. 2a

  1. 55
  2. 63
  3. 104
  4. 3153
  5. 7x7x
  6. ab5b
  7. 633
  8. 2x232x
  9. 36x2y3y
  10. 9ab32b
  11. 9x3y45xy
  12. 27a2b453
  13. 12xy3z452yz
  14. 310+9

  15. 532
  16. 1+2

  17. 5352
  18. 415

  19. 157623
  20. xy

  21. x2+2xy+yx2y
  22. a2ab+bab
  23. 5x+2x254x
  24. x2+3xy+y22xy
  25. 2x+1+2x+12x
  26. x+x21

  27. 56π2π centimeters; 3.45 centimeters

  1. Answer may vary