Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

8.5: Rational Exponents

  • Anonymous
  • LibreTexts

( \newcommand{\kernel}{\mathrm{null}\,}\)

Learning Objectives
  • Write expressions with rational exponents in radical form.
  • Write radical expressions with rational exponents.
  • Perform operations and simplify expressions with rational exponents.
  • Perform operations on radicals with different indices.

Definition of 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, then

xmxn=xm+n

For example, if we have an exponent of 12, then the product rule for exponents implies the following:

512512=512+12=51=5

Here 512 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 213 is one of three equal factors of 2.

213213213=213+13+13=233=21=2

Therefore, 213 is the cube root of 2, and we can write

21/3=32

This is true in general, given any nonzero real number a,

a1/n=na

In other words, the denominator of a fractional exponent determines the index of an nth root.

Example 8.5.1

Rewrite as a radical.

  1. 71/2
  2. 71/3

Solution:

a. 71/2=7

b. 71/3=37

Example 8.5.2

Rewrite as a radical and then simplify.

  1. 811/2
  2. 811/4

Solution:

  1. 811/2=81=9
  2. 811/4=481=434=3
Example 8.5.3

Rewrite as a radical and then simplify.

  1. (125x3)1/3
  2. (32y10)1/5

Solution:

a.

(125x3)1/3=3125x3=353x3=5x

b.

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

523523523=523+23+23=563=52

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

52/3=352

In general, given any real number a,

am/n=nam

An expression with a rational exponent 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 form.

RadicalformExponentialform5x2=x2/5

Example 8.5.4

Rewrite as a radical.

  1. 72/5
  2. 23/4

Solution:

  1. 72/5=572=549
  2. 23/4=423=48
Example 8.5.5

Rewrite as a radical and then simplify

  1. 82/3
  2. (32)3/5

Solution:

a.

82/3=382=364=343=4

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

(32)3/5=5(32)3Replace32with25.=5(25)3Applythepowerruleforexponents.=521515÷5=3,so215=(23)5.=5(23)5Simplify.=23=8

Given a radical expression, we will be asked to find the equivalent in exponential form. Assume all variables are positive.

Example 8.5.6

Rewrite using rational exponents:

3x2

Solution:

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

3x2=x2/3

Answer:

x2/3

Example 8.5.7

Rewrite using rational exponents:

6y3

Solution:

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

6y3=y3/6=y1/2

Answer:

y1/2

It is important to note that the following are equivalent.

nam=(na)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 root:

272/3=3272=3(33)2=336=32=9

Or we can apply the nth root before the power:

272/3=(327)2=(333)2=32=9

The results are the same.

Example 8.5.8

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=(38)2=(2)2=4

Answer:

4

Exercise 8.5.1

Rewrite as a radical and then simplify:

253/2

Answer

125

Some calculators have a caret button ˆ. 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 would type

2(1÷2)=

To calculate 322=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: xmxn=xm+n
Quotient rule: xmxn=xmn,x0
Power rule: (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
Table 8.5.1

These rules allow us to perform operations with rational exponents.

Example 8.5.9

Simplify:

223216

Solution:

223216=223+16Applytheproductrulexmxn=xm+n.=246+16Findequivalentfractionswithacommondenominatorandthenadd.=256

Answer:

256

Example 8.5.10

Simplify:

x1/2x1/3

Solution:

x1/2x1/3=x1213Applythequotientrulexmxn=xmn.=x3626Findequivalentfractionswithacommondenominatorandthensubtract.=x16

Answer:

x16

Example 8.5.11

Simplify:

(y3/4)2/3

Solution:

(y3/4)2/3=y3423Applythepowerrule(xm)n=xmn.=y612Multiplytheexponentsandreduce.=y12

Answer:

y12

Example 8.5.12

Simplify:

(16a4b8)3/4

Solution:

(16a4b8)3/4=(24a4b8)3/4Rewrite16as24.=(24)3/4(a4)3/4(b8)3/4Applythepowerruleforaproduct(xy)n=xnyn.=2434a434b834Applythepowerruletoeachfactor.=23a3b6Simplify.=8a3b6

Answer:

8a3b6

Example 8.5.13

Simplify:

Solution:

Answer:

1125

Exercise 8.5.2

Simplify:

(8a3/4b3)2/3a1/3

Answer

4a1/6b2

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 8.5.14

Multiply:

232

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.

232=212213Equivalentsusingrationalexponents.=212+13Applytheproductruleforexponents.=23+26=256=625

Answer:

625

Example 8.5.15

Divide:

3452

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.

3452=32252=233215Equivalentsusingrationalexponents.=22315Applythequotientruleforexponents.=210315=2715=1527

Answer:

1527

Example 8.5.16

Simplify:

34

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.

34=322=(22/3)1/2Equivalentsusingrationalexponents.=23212Applythepowerruleforexponents.=213=32

Answer:

32

Key Takeaways

  • When converting fractional exponents to 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.
Exercise 8.5.3 Rational Exponents

Express using rational exponents.

  1. 6
  2. 10
  3. 311
  4. 42
  5. 352
  6. 423
  7. 5x
  8. 6x
  9. 6x7
  10. 5x4
Answer

1. 61/2

3. 111/3

5. 52/3

7. x1/5

9. x7/6

Exercise 8.5.4 Rational Exponents

Express in radical form.

  1. 21/2
  2. 51/3
  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
Answer

1. 2

3. 372

5. 4x3

7. 1x

9. 3x

Exercise 8.5.5 Rational Exponents

Write as a radical and then simplify.

  1. 251/2
  2. 361/2
  3. 1211/2
  4. 1441/2
  5. (14)12
  6. (49)12
  7. (4)12
  8. (9)12
  9. (14)12
  10. (116)1/2
  11. 81/3
  12. 1251/3
  13. (127)13
  14. (8125)1/3
  15. (27)13
  16. (64)1/3
  17. 161/4
  18. 6251/4
  19. 811/4
  20. 161/4
  21. 100,0001/5
  22. (32)1/5
  23. (132)15
  24. (1243)1/5
  25. 93/2
  26. 43/2
  27. 85/3
  28. 272/3
  29. 163/2
  30. 322/5
  31. (116)3/4
  32. (181)3/4
  33. (27)2/3
  34. (27)4/3
  35. (32)3/5
  36. (32)4/5
Answer

1. 5

3. 11

5. 12

7. 12

9. 2

11. 2

13. 13

15. −3

17. 2

19. 13

21. 10

23. 12

25. 27

27. 32

29. 64

31. 18

33. 9

35. −8

Exercise 8.5.6 Rational Exponents

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

  1. 23/4
  2. 32/3
  3. 51/5
  4. 71/7
  5. (9)3/2
  6. 93/2
  7. Explain why (4)(3/2) gives an error on a calculator and 4(3/2) gives an answer of −8.
  8. Marcy received a text message from Mark asking her how old she was. In response, Marcy texted back “125(2/3) years old.” Help Mark determine how old Marcy is.
Answer

1. 1.68

3. 1.38

5. Not a real number

7. In the first expression, the square root of a negative number creates an error condition on the calculator. The square root of a negative number is not real. In the second expression, because of the order of operations, the negative sign is applied to the answer after 4 is raised to the (3/2) power.

Exercise 8.5.7 Rational Exponents

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

  1. 22/324/3
  2. 33/231/2
  3. 51/251/3
  4. 21/623/4
  5. y1/4y2/5
  6. x1/2x1/4
  7. 573513
  8. 29/221/2
  9. 2a23a16
  10. 3b12b13
  11. (812)23
  12. (36)2/3
  13. (x23)12
  14. (y34)45
  15. (4x2y4)12
  16. (9x6y2)12
  17. (2x13y23)3
  18. (8x32y12)2
  19. (a34a12)43
  20. (b45b110)103
  21. (4x23y4)12
  22. (27x34y9)13
  23. y12y23y16
  24. x25x12x110
  25. xyx12y13
  26. x54yxy25
  27. 49a57b327a37b14
  28. 16a56b548a12b23
  29. (9x23y6)32x12y
  30. (125x3y35)23xy13
  31. (27a14b32)23a16b12
  32. (25a23b43)32a16b13
Answer

1. 4

3. 55/6

5. y13/20

7. 25

9. 2a1/2

11. 2

13. x1/3

15. 2xy2

17. 8xy2

19 a1/3

21. 2x13y2

23. y

25. x12y23

27. 7a27b54

29. 27x32y8

31. 9b1/2

Exercise 8.5.8 Mixed Indices

Perform the operations.

  1. 3953
  2. 5525
  3. x3x
  4. y4y
  5. 3x24x
  6. 5x33x
  7. 310010
  8. 51634
  9. 3a2a
  10. 5b43b
  11. 3x25x3
  12. 4x33x2
  13. 516
  14. 39
  15. 352
  16. 355
  17. 37
  18. 33
Answer

1. 15313

3. x

5. 12x11

7. 610

9. 6a

11. 15x

13. 54

15. 152

17. 67

Exercise 8.5.9 Discussion Board
  1. Who is credited for devising the notation 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.
Answer

1. Answers may vary


This page titled 8.5: Rational Exponents 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.

Support Center

How can we help?