This is “Rational Exponents”, section 5.5 from the book Advanced Algebra (v. 1.0). For details on it (including licensing), click here.

For more information on the source of this book, or why it is available for free, please see the project's home page. You can browse or download additional books there. You may also download a PDF copy of this book (41 MB) or just this chapter (1 MB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline).

Has this book helped you? Consider passing it on:
Creative Commons supports free culture from music to education. Their licenses helped make this book available to you.
DonorsChoose.org helps people like you help teachers fund their classroom projects, from art supplies to books to calculators.

5.5 Rational Exponents

Learning Objectives

  1. Write expressions with rational exponents in radical form.
  2. Write radical expressions with rational exponents.
  3. Perform operations and simplify expressions with rational exponents.
  4. Perform operations on radicals with different indices.

Rational Exponents

So far, exponents have been limited to integers. In this section, we will define what rational (or fractional) exponents mean and how to work with them. All of the rules for exponents developed up to this point apply. In particular, recall the product rule for exponents. Given any rational numbers m and n, we have xmxn=xm+n For example, if we have an exponent of 1/2, then the product rule for exponents implies the following: 51/251/2=51/2+1/2=51=5 Here 51/2 is one of two equal factors of 5; hence it is a square root of 5, and we can write 51/2=5 Furthermore, we can see that 21/3 is one of three equal factors of 2. 21/321/321/3=21/3+1/3+1/3=23/3=21=2 Therefore, 21/3 is a cube root of 2, and we can write 21/3=23 This is true in general, given any nonzero real number a and integer n2, a1/n=an In other words, the denominator of a fractional exponent determines the index of an nth root.

Example 1

Rewrite as a radical.

  1. 61/2
  2. 61/3

Solution:

  1. 61/2=62=6
  2. 61/3=63

Example 2

Rewrite as a radical and then simplify.

  1. 161/2
  2. 161/4

Solution:

  1. 161/2=16=42=4
  2. 161/4=164=244=2

Example 3

Rewrite as a radical and then simplify.

  1. (64x3)1/3
  2. (32x5y10)1/5

Solution:

a.

(64x3)1/3=64x33=43x33=4x

b.

(32x5y10)1/5=32x5y105=(2)5x5(y2)55=2xy2

Next, consider fractional exponents where the numerator is an integer other than 1. For example, consider the following:

52/352/352/3=52/3+2/3+2/3=56/3=52

This shows that 52/3 is one of three equal factors of 52. In other words, 52/3 is a cube root of 52 and we can write:

52/3=523

In general, given any nonzero real number a where m and n are positive integers (n2),

am/n=amn

An expression with a rational exponentThe fractional exponent m/n that indicates a radical with index n and exponent m: am/n=amn. is equivalent to a radical where the denominator is the index and the numerator is the exponent. Any radical expression can be written with a rational exponent, which we call exponential formAn equivalent expression written using a rational exponent..

RadicalformExponentialformx25=x2/5

Example 4

Rewrite as a radical.

  1. 62/5
  2. 33/4

Solution:

  1. 62/5=625=365
  2. 33/4=334=274

Example 5

Rewrite as a radical and then simplify.

  1. 272/3
  2. (12)5/3

Solution:

We can often avoid very large integers by working with their prime factorization.

a.

272/3=2723=(33)23Replace27with33.=363Simplify.=32=9

b.

(12)5/3=(12)53Replace12with223.=(223)53Applytherulesforexponents.=210353Simplify.=29233323=2332323=24183

Given a radical expression, we might want to find the equivalent in exponential form. Assume all variables are positive.

Example 6

Rewrite using rational exponents: x35.

Solution:

Here the index is 5 and the power is 3. We can write

x35=x3/5

Answer: x3/5

Example 7

Rewrite using rational exponents: y36.

Solution:

Here the index is 6 and the power is 3. We can write

y36=y3/6=y1/2

Answer: y1/2

It is important to note that the following are equivalent.

am/n=amn=(an)m

In other words, it does not matter if we apply the power first or the root first. For example, we can apply the power before the nth root:

272/3=2723=(33)23=363=32=9

Or we can apply the nth root before the power:

272/3=(273)2=(333)2=(3)2=9

The results are the same.

Example 8

Rewrite as a radical and then simplify: (8)2/3.

Solution:

Here the index is 3 and the power is 2. We can write

(8)2/3=(83)2=(2)2=4

Answer: 4

Try this! Rewrite as a radical and then simplify: 1003/2.

Answer: 1,000

Some calculators have a caret button ^ which is used for entering exponents. If so, we can calculate approximations for radicals using it and rational exponents. For example, to calculate 2=21/2=2^(1/2)1.414, we make use of the parenthesis buttons and type

2^(1÷2)=

To calculate 223=22/3=2^(2/3)1.587, we would type

2^(2÷3)=

Operations Using the Rules of Exponents

In this section, we review all of the rules of exponents, which extend to include rational exponents. If given any rational numbers m and n, then we have

Product rule for exponents:

xmxn=xm+n

Quotient rule for exponents:

xmxn=xmn,x0

Power rule for exponents:

(xm)n=xmn

Power rule for a product:

(xy)n=xnyn

Power rule for a quotient:

(xy)n=xnyn,y0

Negative exponents:

xn=1xn

Zero exponent:

x0=1,x0

These rules allow us to perform operations with rational exponents.

Example 9

Simplify: 71/374/9.

Solution:

71/374/9=71/3+4/9Applytheproductrulexmxn=xm+n.=73/9+4/9=77/9

Answer: 77/9

Example 10

Simplify: x3/2x2/3.

Solution:

x3/2x2/3=x3/22/3Applythequotientrulexmxn=xmn.=x9/64/6=x5/6

Answer: x5/6

Example 11

Simplify: (y3/4)2/3.

Solution:

(y3/4)2/3=y(3/4)(2/3)Applythepowerrule(xm)n=xmn.=y6/12Multiplytheexponentsandreduce.=y1/2

Answer: y1/2

Example 12

Simplify: (81a8b12)3/4.

Solution:

(81a8b12)3/4=(34a8b12)3/4Rewrite81as34.=(34)3/4(a8)3/4(b12)3/4Applythepowerruleforaproduct(xy)n=xnyn.=34(3/4)a8(3/4)b12(3/4)Applythepowerruletoeachfactor.=33a6b9Simplify.=27a6b9

Answer: 27a6b9

Example 13

Simplify: (9x4)3/2.

Solution:

(9x4)3/2=1(9x4)3/2Applythedefinitionofnegativeexponentsxn=1xn.=1(32x4)3/2Write9as32andapplytherulesofexponents.=132(3/2)x4(3/2)=133x6=127x6

Answer: 127x6

Try this! Simplify: (125a1/4b6)2/3a1/6.

Answer: 25b4

Radical Expressions with Different Indices

To apply the product or quotient rule for radicals, the indices of the radicals involved must be the same. If the indices are different, then first rewrite the radicals in exponential form and then apply the rules for exponents.

Example 14

Multiply: 223.

Solution:

In this example, the index of each radical factor is different. Hence the product rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents. Then apply the product rule for exponents.

223=21/221/3Equivalentsusingrationalexponents=21/2+1/3Applytheproductruleforexponents.=25/6=256

Answer: 256

Example 15

Divide: 4325.

Solution:

In this example, the index of the radical in the numerator is different from the index of the radical in the denominator. Hence the quotient rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents and then apply the quotient rule for exponents.

4325=22325=22/321/5Equivalentsusingrationalexponents=22/31/5Applythequotientruleforexponents.=27/15=2715

Answer: 2715

Example 16

Simplify: 43.

Solution:

Here the radicand of the square root is a cube root. After rewriting this expression using rational exponents, we will see that the power rule for exponents applies.

43=223=(22/3)1/2Equivalentsusingrationalexponents=2(2/3)(1/2)Applythepowerruleforexponents.=21/3=23

Answer: 23

Key Takeaways

  • Any radical expression can be written in exponential form: amn=am/n.
  • Fractional exponents indicate radicals. Use the numerator as the power and the denominator as the index of the radical.
  • All the rules of exponents apply to expressions with rational exponents.
  • If operations are to be applied to radicals with different indices, first rewrite the radicals in exponential form and then apply the rules for exponents.

Topic Exercises

    Part A: Rational Exponents

      Express using rational exponents.

    1. 10

    2. 6

    3. 33

    4. 54

    5. 523

    6. 234

    7. 493

    8. 93

    9. x5

    10. x6

    11. x76

    12. x45

    13. 1x
    14. 1x23

      Express in radical form.

    1. 101/2

    2. 111/3

    3. 72/3

    4. 23/5

    5. x3/4

    6. x5/6

    7. x1/2

    8. x3/4

    9. (1x)1/3
    10. (1x)3/5
    11. (2x+1)2/3

    12. (5x1)1/2

      Write as a radical and then simplify.

    1. 641/2

    2. 491/2

    3. (14)1/2
    4. (49)1/2
    5. 41/2

    6. 91/2

    7. (14)1/2
    8. (116)1/2
    9. 81/3

    10. 1251/3

    11. (127)1/3
    12. (8125)1/3
    13. (27)1/3

    14. (64)1/3

    15. 161/4

    16. 6251/4

    17. 811/4

    18. 161/4

    19. 100,0001/5

    20. (32)1/5

    21. (132)1/5
    22. (1243)1/5
    23. 93/2

    24. 43/2

    25. 85/3

    26. 272/3

    27. 163/2

    28. 322/5

    29. (116)3/4
    30. (181)3/4
    31. (27)2/3

    32. (27)4/3

    33. (32)3/5

    34. (32)4/5

      Use a calculator to approximate an answer rounded to the nearest hundredth.

    1. 21/2

    2. 21/3

    3. 23/4

    4. 32/3

    5. 51/5

    6. 71/7

    7. (9)3/2

    8. 93/2

    9. Explain why (−4)^(3/2) gives an error on a calculator and −4^(3/2) gives an answer of −8.

    10. Marcy received a text message from Mark asking her age. In response, Marcy texted back “125^(2/3) years old.” Help Mark determine Marcy’s age.

    Part B: Operations Using the Rules of Exponents

      Perform the operations and simplify. Leave answers in exponential form.

    1. 53/251/2

    2. 32/337/3

    3. 51/251/3

    4. 21/623/4

    5. y1/4y2/5

    6. x1/2x1/4

    7. 511/352/3
    8. 29/221/2
    9. 2a2/3a1/6
    10. 3b1/2b1/3
    11. (81/2)2/3
    12. (36)2/3
    13. (x2/3)1/2
    14. (y3/4)4/5
    15. (y8)1/2
    16. (y6)2/3
    17. (4x2y4)1/2
    18. (9x6y2)1/2
    19. (2x1/3y2/3)3
    20. (8x3/2y1/2)2
    21. (36x4y2)1/2
    22. (8x3y6z3)1/3
    23. (a3/4a1/2)4/3
    24. (b4/5b1/10)10/3
    25. (4x2/3y4)1/2
    26. (27x3/4y9)1/3
    27. y1/2y2/3y1/6
    28. x2/5x1/2x1/10
    29. xyx1/2y1/3
    30. x5/4yxy2/5
    31. 49a5/7b3/27a3/7b1/4
    32. 16a5/6b5/48a1/2b2/3
    33. (9x2/3y6)3/2x1/2y
    34. (125x3y3/5)2/3xy1/3
    35. (27a1/4b3/2)2/3a1/6b1/2
    36. (25a2/3b4/3)3/2a1/6b1/3
    37. (16x2y1/3z2/3)3/2

    38. (81x8y4/3z4)3/4

    39. (100a2/3b4c3/2)1/2

    40. (125a9b3/4c1)1/3

    Part C: Radical Expressions with Different Indices

      Perform the operations.

    1. 9335

    2. 5255

    3. xx3

    4. yy4

    5. x23x4

    6. x35x3

    7. 100310
    8. 16543
    9. a23a
    10. b45b3
    11. x23x35
    12. x34x23
    13. 165

    14. 93

    15. 253

    16. 553

    17. 73

    18. 33

    Part D: Discussion Board

    1. Who is credited for devising the notation that allows for rational exponents? What are some of his other accomplishments?

    2. When using text, it is best to communicate nth roots using rational exponents. Give an example.

Answers

  1. 101/2

  2. 31/3

  3. 52/3

  4. 72/3

  5. x1/5

  6. x7/6

  7. x1/2

  8. 10

  9. 493

  10. x34

  11. 1x
  12. x3

  13. (2x+1)23

  14. 8

  15. 12

  16. 12

  17. 2

  18. 2

  19. 13

  20. −3

  21. 2

  22. 13

  23. 10

  24. 12

  25. 27

  26. 32

  27. 64

  28. 18

  29. 9

  30. −8

  31. 1.41

  32. 1.68

  33. 1.38

  34. Not a real number

  35. Answer may vary

  1. 25

  2. 55/6

  3. y13/20

  4. 125

  5. 2a1/2

  6. 2

  7. x1/3

  8. 1y4
  9. 2xy2

  10. 8xy2

  11. 16x2y
  12. a1/3

  13. 2x1/3y2
  14. y

  15. x1/2y2/3

  16. 7a2/7b5/4

  17. 27x1/2y8

  18. 9b1/2

  19. y1/264x3z
  20. a1/3b3/410b2
  1. 31315

  2. x56

  3. x1112

  4. 106

  5. a6

  6. x15

  7. 45

  8. 215

  9. 76

  1. Answer may vary