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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.
- 61/2
- 61/3
Solution:
- 61/2=62=6
- 61/3=6 3
Example 2
Rewrite as a radical and then simplify.
- 161/2
- 161/4
Solution:
- 161/2=16=42 =4
- 161/4=164=244=2
Example 3
Rewrite as a radical and then simplify.
- (64x3)1/3
- (−32x5y10)1/5
Solution:
a.
(64x3 )1/3=64x33=43x33 =4x
b.
(−32x5y10)1/ 5=−32x5y105=(−2)5 x5(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..
Radicalform Exponentialformx2 5=x2/5
Example 4
Rewrite as a radical.
- 62/5
- 33/4
Solution:
- 62/5=625=365
- 33/4 =334=274
Example 5
Rewrite as a radical and then simplify.
- 272/3
- (12)5/3
Solution:
We can often avoid very large integers by working with their prime factorization.
a.
272/3=2723 =(33)23 Replace27with33.=363 Simplify.=32 =9
b.
( 12)5/3=(12)53Repl ace12with22⋅3.=(22⋅3)5 3Applytherulesforexponents. =210⋅353Simpl ify.=29⋅2⋅33⋅323=23⋅3⋅2⋅323 =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)2 3=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+4/9Applytheproductrulexm⋅xn=xm+n. =73/9+4/9=77/9
Answer: 77/9
Example 10
Simplify: x3/2x2 /3.
Solution:
x3/2x2/3=x 3/2−2/3Applythequotientrulexmxn =xm−n.=x9/6−4/6=x5/6
Answer: x5/6
Example 11
Simplify: (y3/4)2/3.
Solution:
(y 3/4)2/3=y(3/4)(2/3)Applyt hepowerrule(xm)n=xm⋅n.= 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)3/4A pplythepowerruleforaproduct(xy)n=x nyn.=34(3/4)a8(3/4)b12( 3/4)Applythepowerruletoeachfact or.=33a6b9Simplify.= 27a6b9
Answer: 27a6b9
Example 13
Simplify: (9x4)−3/2.
Solution:
(9x4)−3/2=1(9x4)3/2 Applythedefinitionofnegativeexponen tsx−n=1xn.=1(32x4) 3/2Write9as32andapplyth erulesofexponents.=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⋅2 3.
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⋅21/3Equivalentsusingrationalexponents=21/2+1/3Applytheproduc truleforexponents.=25/6=256
Answer: 256
Example 15
Divide: 43 25.
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=2 2325=22/3 21/5Equivalentsusingrationale xponents=22/3−1/5App lythequotientruleforexponents.=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)Applythepowerr uleforexponents.=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
10
6
33
54
523
234
493
93
x5
x6
x76
x45
1x
1x23
101/2
111/3
72/3
23/5
x3/4
x5/6
x−1/2
x−3/4
(1x)−1/3
(1x) −3/5
(2x+1)2/3
(5x−1)1/2
641/2
491/2
(14)1/2
(49)1 /2
4−1/2
9− 1/2
(14)−1/2
(116)−1/2
81/3
1251/3
(127)1/3
(8125)1/3
(− 27)1/3
(−64)1/3
161/4
6251/4
81−1/4
16−1/4
100,0001/5
(− 32)1/5
(132)1/5
(1243)1/5
93/2
43/2
85/3
272/3
163/2
322/5
(116)3/ 4
(181)3/4
(−27)2/3
(−27)4/3
(−32)3/5
(− 32)4/5
21/ 2
21/3
23/4
32/3
51/5
71/7
(−9)3 /2
−93/2
Explain why (−4)^(3/2) gives an error on a calculator and −4^(3/2) gives an answer of −8.
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 A: Rational Exponents
Express using rational exponents.
Express in radical form.
Write as a radical and then simplify.
Use a calculator to approximate an answer rounded to the nearest hundredth.
53/2⋅51/2
3 2/3⋅37/3
51/2⋅51/3
21/6⋅23/4
y1/4 ⋅y2/5
x1/2⋅x1/4
511/352/3
29/221/ 2
2a2/3a1/6
3b1/2b1/3
(81/2)2/3
(36)2/3
(x2/ 3)1/2
(y3/4)4/5
(y8)−1/2
(y6)−2/ 3
(4x2y4)1/2
(9x6y2)1/2
(2x1/3y2/3 )3
(8x3/2y1/2)2
(36x4y2)−1/2
(8x3y6 z−3)−1/3
(a3/4a1/2 )4/3
(b4/5b1/10)10/ 3
(4x2/3y4)1/2
(27x3/4y9)1/3
y1 /2y2/3y1/6
x2/5x1/2 x1/10
xyx1/2y1/3
x5/4yxy2/5
49a5/7b3 /27a3/7b1/4
16a5/6b5/4 8a1/2b2/3
(9x2/3y6) 3/2x1/2y
(125x3y3/5)2 /3xy1/3
(27a1/4b3/2) 2/3a1/6b1/2
(25a2/3b4 /3)3/2a1/6b1/3
(16 x2y−1/3z2/3)−3/2
(81x 8y−4/3z−4)−3/4
(100a −2/3b4c−3/2)−1/2
(125 a9b−3/4c−1)−1/3
Part B: Operations Using the Rules of Exponents
Perform the operations and simplify. Leave answers in exponential form.
93⋅35
5⋅255
x⋅x3
y⋅y4
x23 ⋅x4
x35⋅x3
100310
16543
a23a
b45b3
x23x35
x34 x23
165
93
253
553
73
33
Part C: Radical Expressions with Different Indices
Perform the operations.
Who is credited for devising the notation that allows for rational exponents? What are some of his other accomplishments?
When using text, it is best to communicate nth roots using rational exponents. Give an example.
Part D: Discussion Board
Answers
101/2
31/3
52/3
72/3
x1/5
x7 /6
x−1/2
10
493
x34
1x
x 3
(2x+1)23
8
12
12
2
2
13
−3
2
13
10
12
27
32
64
18
9
−8
1.41
1.68
1.38
Not a real number
Answer may vary
25
55/6
y13/20
125
2a1/2
2
x1/3
1y4
2xy2
8xy2
16x2y
a1/3
2x1/3y2
y
x1/2y2/3
7a2/7b5/4
27x1/2y8
9b1/2
y1/264x3z
a1/3b3/410b2
31315
x5 6
x1112
106
a6
x15
45
2 15
76
Answer may vary