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## 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 $xm⋅xn=xm+n$ For example, if we have an exponent of 1/2, then the product rule for exponents implies the following: $51/2⋅51/2=51/2+1/2=51=5$ 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. $21/3⋅21/3⋅21/3=21/3+1/3+1/3=23/3=21=2$ 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 $n≥2$, $a1/n=an$ 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:

$52/3⋅52/3⋅52/3=52/3+2/3+2/3=56/3=52$

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 ($n≥2$),

$am/n=amn$

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..

$Radical formExponential formx25=x2/5$

### 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.

$272/3=2723=(33)23Replace 27 with 33.=363Simplify.=32=9$

b.

$(12)5/3=(12)53Replace 12 with 22⋅3.=(22⋅3)53Apply the rules for exponents.=210⋅353Simplify.=29⋅2⋅33⋅323=23⋅3⋅2⋅323=24 183$

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: $xm⋅xn=xm+n$ Quotient rule for exponents: $xmxn=xm−n, x≠0$ Power rule for exponents: $(xm)n=xm⋅n$ Power rule for a product: $(xy)n=xnyn$ Power rule for a quotient: $(xy)n=xnyn , y≠0$ Negative exponents: $x−n=1xn$ Zero exponent: $x0=1, x≠0$

These rules allow us to perform operations with rational exponents.

### Example 9

Simplify: $71/3⋅74/9.$

Solution:

$71/3⋅74/9=71/3+4/9Apply the product rule xm⋅xn=xm+n.=73/9+4/9=77/9$

Answer: $77/9$

### Example 10

Simplify: $x3/2x2/3$.

Solution:

$x3/2x2/3=x3/2−2/3Apply the quotient rule xmxn=xm−n .=x9/6−4/6=x5/6$

Answer: $x5/6$

### Example 11

Simplify: $(y3/4)2/3.$

Solution:

$(y3/4)2/3=y(3/4)(2/3)Apply the power rule (xm)n=xm⋅n .=y6/12 Multiply the exponents and reduce.=y1/2$

Answer: $y1/2$

### Example 12

Simplify: $(81a8b12)3/4$.

Solution:

$(81a8b12)3/4=(34a8b12)3/4Rewrite 81 as 34.=(34)3/4(a8)3/4(b12)3/4Apply the power rule for a product (xy)n=xnyn .=34(3/4)a8(3/4)b12(3/4)Apply the power rule to each factor.=33a6b9Simplify.=27a6b9$

Answer: $27a6b9$

### Example 13

Simplify: $(9x4)−3/2.$

Solution:

$(9x4)−3/2=1(9x4)3/2Apply the definition of negative exponents x−n=1xn.=1(32x4)3/2Write 9 as 32 and apply the rules of exponents.=132(3/2)x4(3/2)=133⋅x6=127x6$

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: $2⋅23.$

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.

$2⋅23=21/2⋅21/3Equivalents using rational exponents=21/2+1/3Apply the product rule for exponents.=25/6=256$

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.

$4325=22325=22/321/5Equivalents using rational exponents=22/3−1/5Apply the quotient rule for exponents.=27/15=2715$

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.

$43 = 223 =(22/3)1/2Equivalents using rational exponents=2(2/3)(1/2)Apply the power rule for exponents.=21/3=23$

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.

### 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. $x−1/2$

8. $x−3/4$

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

12. $(5x−1)1/2$

Write as a radical and then simplify.

1. $641/2$

2. $491/2$

3. $(14)1/2$
4. $(49)1/2$
5. $4−1/2$

6. $9−1/2$

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. $81−1/4$

18. $16−1/4$

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. $53/2⋅51/2$

2. $32/3⋅37/3$

3. $51/2⋅51/3$

4. $21/6⋅23/4$

5. $y1/4⋅y2/5$

6. $x1/2⋅x1/4$

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. $(8x3y6z−3)−1/3$
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. $(16x2y−1/3z2/3)−3/2$

38. $(81x8y−4/3z−4)−3/4$

39. $(100a−2/3b4c−3/2)−1/2$

40. $(125a9b−3/4c−1)−1/3$

### Part C: Radical Expressions with Different Indices

Perform the operations.

1. $93⋅35$

2. $5⋅255$

3. $x⋅x3$

4. $y⋅y4$

5. $x23⋅x4$

6. $x35⋅x3$

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. $x−1/2$

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