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5.2 Simplifying Radical Expressions

Learning Objectives

  1. Simplify radical expressions using the product and quotient rule for radicals.
  2. Use formulas involving radicals.

Simplifying Radical Expressions

An algebraic expression that contains radicals is called a radical expressionAn algebraic expression that contains radicals.. We use the product and quotient rules to simplify them.

Example 1

Simplify: 27x33.

Solution:

Use the fact that ann=a when n is odd.

27x33=33x33Applytheproductruleforradicals.=333x33Simplify.=3x=3x

Answer: 3x

Example 2

Simplify: 16y44.

Solution:

Use the fact that ann=|a| when n is even.

16y44=24y44Applytheproductruleforradicals.=244y44Simplify.=2|y|=2|y|

Since y is a variable, it may represent a negative number. Thus we need to ensure that the result is positive by including the absolute value.

Answer: 2|y|

Important Note

Typically, at this point in algebra we note that all variables are assumed to be positive. If this is the case, then y in the previous example is positive and the absolute value operator is not needed. The example can be simplified as follows.

16y44=24y44=244y44=2y

In this section, we will assume that all variables are positive. This allows us to focus on calculating nth roots without the technicalities associated with the principal nth root problem. For this reason, we will use the following property for the rest of the section,

ann=a,ifa0         nthroot

When simplifying radical expressions, look for factors with powers that match the index.

Example 3

Simplify: 12x6y3.

Solution:

Begin by determining the square factors of 12, x6, and y3.

12=223x6=(x3)2y3=y2y}Squarefactors

Make these substitutions, and then apply the product rule for radicals and simplify.

12x6y3=223(x3)2y2yApplytheproductruleforradicals.=22(x3)2y23ySimplify.=2x3y3y=2x3y3y

Answer: 2x3y3y

Example 4

Simplify: 18a5b8.

Solution:

Begin by determining the square factors of 18, a5, and b8.

18=232a5=a2a2a=(a2)2ab8=b4b4=(b4)2}Squarefactors

Make these substitutions, apply the product and quotient rules for radicals, and then simplify.

18a5b8=232(a2)2a(b4)2Applytheproductandquotientruleforradicals.=32(a2)22a(b4)2Simplify.=3a22ab4

Answer: 3a22ab4

Example 5

Simplify: 80x5y73.

Solution:

Begin by determining the cubic factors of 80, x5, and y7.

80=245=2325x5=x3x2y7=y6y=(y2)3y}Cubicfactors

Make these substitutions, and then apply the product rule for radicals and simplify.

80x5y73=2325x3x2(y2)3y3=233x33(y2)3325x2y3=2xy210x2y3=2xy210x2y3

Answer: 2xy210x2y3

Example 6

Simplify 9x6y3z93.

Solution:

The coefficient 9=32, and thus does not have any perfect cube factors. It will be left as the only remaining radicand because all of the other factors are cubes, as illustrated below:

x6=(x2)3y3=(y)3z9=(z3)3}Cubicfactors

Replace the variables with these equivalents, apply the product and quotient rules for radicals, and then simplify.

9x6y3z93=9(x2)3y3(z3)33=93(x2)33y33(z3)33=93x2yz3=x293yz3

Answer: x293yz3

Example 7

Simplify: 81a4b54.

Solution:

Determine all factors that can be written as perfect powers of 4. Here, it is important to see that b5=b4b. Hence the factor b will be left inside the radical.

81a4b54=34a4b4b4=344a44b44b4=3abb4=3abb4

Answer: 3abb4

Example 8

Simplify: 32x3y6z55.

Solution:

Notice that the variable factor x cannot be written as a power of 5 and thus will be left inside the radical. In addition, y6=y5y; the factor y will be left inside the radical as well.

32x3y6z55=(2)5x3y5yz55=(2)55y55z55x3y5=2yzx3y5=2yzx3y5

Answer: 2yzx3y5

Tip: To simplify finding an nth root, divide the powers by the index.

a6=a3,     which is  a6÷2=a3b63=b2,     which is  b6÷3=b2c66=c ,       which is   c6÷6=c1

If the index does not divide into the power evenly, then we can use the quotient and remainder to simplify. For example,

a5=a2a,      which is  a5÷2=a2r1b53=bb23,       which is   b5÷3=b1r2c145=c2c45,     which is  c14÷5=c2r4

The quotient is the exponent of the factor outside of the radical, and the remainder is the exponent of the factor left inside the radical.

Try this! Simplify: 162a7b5c43.

Answer: 3a2bc6ab2c3

Formulas Involving Radicals

Formulas often consist of radical expressions. For example, the period of a pendulum, or the time it takes a pendulum to swing from one side to the other and back, depends on its length according to the following formula.

T=2πL32

Here T represents the period in seconds and L represents the length in feet of the pendulum.

Example 9

If the length of a pendulum measures 112 feet, then calculate the period rounded to the nearest tenth of a second.

Solution:

Substitute 112=32 for L and then simplify.

T=2πL32=2π3232=2π32132Applythequotientruleforradicals.=2π364Simplify.=2π38=π341.36

Answer: The period is approximately 1.36 seconds.

Frequently you need to calculate the distance between two points in a plane. To do this, form a right triangle using the two points as vertices of the triangle and then apply the Pythagorean theorem. Recall that the Pythagorean theorem states that if given any right triangle with legs measuring a and b units, then the square of the measure of the hypotenuse c is equal to the sum of the squares of the legs: a2+b2=c2. In other words, the hypotenuse of any right triangle is equal to the square root of the sum of the squares of its legs.

Example 10

Find the distance between (−5, 3) and (1, 1).

Solution:

Form a right triangle by drawing horizontal and vertical lines though the two points. This creates a right triangle as shown below:

The length of leg b is calculated by finding the distance between the x-values of the given points, and the length of leg a is calculated by finding the distance between the given y-values.

a=31=2 unitsb=1(5)=1+5=6 units

Next, use the Pythagorean theorem to find the length of the hypotenuse.

c=22+62=4+36=40=410=210 units

Answer: The distance between the two points is 210 units.

Generalize this process to produce a formula that can be used to algebraically calculate the distance between any two given points.

Given two points, (x1,y1) and (x2,y2), the distance, d, between them is given by the distance formulaGiven two points (x1,y1) and (x2,y2), calculate the distance d between them using the formula d=(x2x1)2+(y2y1)2., d=(x2x1)2+(y2y1)2.

Example 11

Calculate the distance between (−4, 7) and (2, 1).

Solution:

Use the distance formula with the following points.

(x1,y1)(x2,y2)(4,7)(2,1)

It is a good practice to include the formula in its general form before substituting values for the variables; this improves readability and reduces the probability of making errors.

d=(x2x1)2+(y2y1)2=(2(4))2+(17)2=(2+4)2+(17)2=(6)2+(6)2=36+36=72=362=62

Answer: The distance between the two points is 62 units.

Example 12

Do the three points (2, −1), (3, 2), and (8, −3) form a right triangle?

Solution:

The Pythagorean theorem states that having side lengths that satisfy the property a2+b2=c2 is a necessary and sufficient condition of right triangles. In other words, if you can show that the sum of the squares of the leg lengths of the triangle is equal to the square of the length of the hypotenuse, then the triangle must be a right triangle. First, calculate the length of each side using the distance formula.

Geometry

Calculation

Points: (2, −1) and (8, −3)

a=(82)2+[3(1)]2=(6)2+(3+1)2=36+(2)2=36+4=40=210

Points: (2, −1) and (3, 2)

b=(32)2+[2(1)]2=(1)2+(2+1)2=1+(3)2=1+9=10

Points: (3, 2) and (8, −3)

c=(83)2+(32)2=(5)2+(5)2=25+25=50=52

Now we check to see if a2+b2=c2.

a2+b2=c2(210)2+(10)2=(52)24(10)2+(10)2=25(2)2410+10=25250=50

Answer: Yes, the three points form a right triangle.

Try this! The speed of a vehicle before the brakes were applied can be estimated by the length of the skid marks left on the road. On wet concrete, the speed v in miles per hour can be estimated by the formula v=23d, where d represents the length of the skid marks in feet. Estimate the speed of a vehicle before applying the brakes if the skid marks left behind measure 27 feet. Round to the nearest mile per hour.

Answer: 18 miles per hour

Key Takeaways

  • To simplify a radical expression, look for factors of the radicand with powers that match the index. If found, they can be simplified by applying the product and quotient rules for radicals, as well as the property ann=a, where a is nonnegative.
  • A radical expression is simplified if its radicand does not contain any factors that can be written as perfect powers of the index.
  • We typically assume that all variable expressions within the radical are nonnegative. This allows us to focus on simplifying radicals without the technical issues associated with the principal nth root. If this assumption is not made, we will ensure a positive result by using absolute values when simplifying radicals with even indices.

Topic Exercises

    Part A: Simplifying Radical Expressions

      Assume that the variable could represent any real number and then simplify.

    1. 9x2

    2. 16y2

    3. 8y33

    4. 125a33

    5. 64x44

    6. 81y44

    7. 36a4

    8. 100a8

    9. 4a6

    10. a10

    11. 18a4b5

    12. 48a5b3

    13. 128x6y86

    14. a6b7c86

    15. (5x4)2

    16. (3x5)4

    17. x26x+9

    18. x210x+25

    19. 4x2+12x+9

    20. 9x2+6x+1

      Simplify. (Assume all variable expressions represent positive numbers.)

    1. 49a2

    2. 64b2

    3. x2y2

    4. 25x2y2z2

    5. 180x3

    6. 150y3

    7. 49a3b2

    8. 4a4b3c

    9. 45x5y3

    10. 50x6y4

    11. 64r2s6t5

    12. 144r8s6t2

    13. (x+1)2

    14. (2x+3)2

    15. 4(3x1)2

    16. 9(2x+3)2

    17. 9x325y2
    18. 4x59y4
    19. m736n4
    20. 147m9n6
    21. 2r2s525t4
    22. 36r5s2t6
    23. 27a33

    24. 125b33

    25. 250x4y33

    26. 162a3b53

    27. 64x3y6z93

    28. 216x12y33

    29. 8x3y43

    30. 27x5y33

    31. a4b5c63

    32. a7b5c33

    33. 8x427y33
    34. x5125y63
    35. 360r5s12t133

    36. 540r3s2t93

    37. 81x44

    38. x4y44

    39. 16x4y84

    40. 81x12y44

    41. a4b5c64

    42. 54a6c84

    43. 128x64

    44. 243y74

    45. 32m10n55
    46. 37m9n105
    47. 34x2

    48. 79y2

    49. 5x4x2y

    50. 3y16x3y2

    51. 12aba5b3

    52. 6a2b9a7b2

    53. 2x8x63

    54. 5x227x33

    55. 2ab8a4b53

    56. 5a2b27a3b33

      Rewrite the following as a radical expression with coefficient 1.

    1. 3x6x

    2. 5y5y

    3. ab10a

    4. 2ab2a

    5. m2nmn

    6. 2m2n33n

    7. 2x3x3

    8. 3yy23

    9. 2y24y4

    10. x2y9xy25

    Part B: Formulas Involving Radicals

      The period T in seconds of a pendulum is given by the formula T=2πL32 where L represents the length in feet of the pendulum. Calculate the period, given each of the following lengths. Give the exact value and the approximate value rounded to the nearest tenth of a second.

    1. 8 feet

    2. 32 feet

    3. 12 foot

    4. 18 foot

      The time t in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet the object has fallen. Calculate the time it takes an object to fall, given each of the following distances. Give the exact value and the approximate value rounded to the nearest tenth of a second.

    1. 48 feet

    2. 80 feet

    3. 192 feet

    4. 288 feet

    5. The speed of a vehicle before the brakes were applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed v in miles per hour can be estimated by the formula v=26d, where d represents the length of the skid marks in feet. Estimate the speed of a vehicle before applying the brakes on dry pavement if the skid marks left behind measure 27 feet. Round to the nearest mile per hour.

    6. The radius r of a sphere can be calculated using the formula r=6π2V32π, where V represents the sphere’s volume. What is the radius of a sphere if the volume is 36π cubic centimeters?

      Given the function find the y-intercept

    1. f(x)=x+12

    2. f(x)=x+83

    3. f(x)=x83

    4. f(x)=x+273

    5. f(x)=x+163

    6. f(x)=x+331

      Use the distance formula to calculate the distance between the given two points.

    1. (5, −7) and (3, −8)

    2. (−9, 7) and (−8, 4)

    3. (−3, −4) and (3, −6)

    4. (−5, −2) and (1, −6)

    5. (−1, 1) and (−4, 10)

    6. (8, −3) and (2, −12)

    7. (0, −6) and (−3, 0)

    8. (0, 0) and (8, −4)

    9. (12,12) and (1,32)

    10. (13,2) and (53,23)

      Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.

    1. (2,−1), (−1,2), and (6,3)

    2. (−5,2), (−1, −2), and (−2,5)

    3. (−5,0), (0,3), and (6,−1)

    4. (−4,−1), (−2,5), and (7,2)

    5. (1,−2), (2,3), and (−3,4)

    6. (−2,1), (−1,−1), and (1,3)

    7. (−4,0), (−2,−10), and (3,−9)

    8. (0,0), (2,4), and (−2,6)

    Part D: Discussion Board

    1. Give a value for x such that x2x. Explain why it is important to assume that the variables represent nonnegative numbers.

    2. Research and discuss the accomplishments of Christoph Rudolff. What is he credited for?

    3. What is a surd, and where does the word come from?

    4. Research ways in which police investigators can determine the speed of a vehicle after an accident has occurred. Share your findings on the discussion board.

Answers

  1. 3|x|

  2. 2y

  3. 2|x|

  4. 6a2

  5. 2|a3|
  6. 3a2b22b

  7. 2|xy|2y26

  8. |5x4|

  9. |x3|

  10. |2x+3|

  11. 7a

  12. xy

  13. 6x5x

  14. 7aba

  15. 3x2y5xy

  16. 8rs3t2t

  17. x+1

  18. 2(3x1)

  19. 3xx5y
  20. m3m6n2
  21. rs22s5t2
  22. 3a

  23. 5xy2x3

  24. 4xy2z3

  25. 2xyy3

  26. abc2ab23

  27. 2xx33y
  28. 2rs4t445r2t3

  29. 3x

  30. 2xy2

  31. abcbc24

  32. 2x8x24

  33. 2m2n
  34. 6x

  35. 10x2y

  36. 12a3b2ab

  37. 4x3

  38. 4a2b2ab23

  39. 54x3

  40. 10a3b2

  41. m5n3

  42. 24x43

  43. 64y94

  1. π seconds; 3.1 seconds

  2. π4 seconds; 0.8 seconds

  3. 3 seconds; 1.7 seconds

  4. 23 seconds; 3.5 seconds

  5. 25 miles per hour

  6. (0,23)

  7. (0,2)

  8. (0,223)
  9. 5 units

  10. 210 units

  11. 310 units

  12. 35 units

  13. 52 units

  14. Right triangle

  15. Not a right triangle

  16. Right triangle

  17. Right triangle

  1. Answer may vary

  2. Answer may vary