NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-6 Study Guide and Intervention
Rational Exponents
Rational Exponents and Radicals
1
𝟏
𝑛
For any real number b and any positive integer n, 𝑏 𝑛 = √𝑏, except when b < 0 and n is even.
Definition of 𝒃𝒏
𝒎
𝑚
Definition of 𝒃 𝒏
𝑛
𝑛
𝑚
For any nonzero real number b, and any integers m and n, with n > 1, 𝑏 𝑛 = √𝑏𝑚 = ( √𝑏 ) ,
except when b < 0 and n is even.
𝟏
𝟏
𝟑
Example 1: Write 𝟐𝟖𝟐 in radical form.
Example 2: Evaluate (
Notice that 28 > 0.
Notice that –8 < 0, –125 < 0, and 3 is odd.
1
2
28 = √28
(
= √22 ⋅ 7
−8
−125
1
3
) =3
−𝟖
−𝟏𝟐𝟓
).
3
√ −8
√−125
−2
= √22 ⋅ √7
=
= 2√7
=5
−5
2
Exercises
Write each expression in radical form, or write each radical in exponential form.
1
1
3
1. 117
2. 153
3. 3002
4. √47
5. √3𝑎5 𝑏2
3
4
6. √162𝑝5
Evaluate each expression.
2
7. −273
Chapter 6
1
1
8. 2163
9. (0.0004)2
38
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-6 Study Guide and Intervention (continued)
Rational Exponents
Simplify Expressions All the properties of powers from Lesson 6-1 apply to rational exponents. When you simplify
expressions with rational exponents, leave the exponent in rational form, and write the expression with all positive
exponents. Any exponents in the denominator must be positive integers.
When you simplify radical expressions, you may use rational exponents to simplify, but your answer should be in radical
form. Use the smallest index possible.
𝟐
𝟑
Example 1: Simplify 𝒚𝟑 ⋅ 𝒚𝟖.
2
3
2
3
𝟒
Example 2: Simplify √𝟏𝟒𝟒𝒙𝟔.
25
𝑦 3 ⋅ 𝑦 8 = 𝑦 3 + 8 = 𝑦 24
1
4
√144𝑥 6 = (144𝑥 6 )4
1
= (24 ⋅ 32 ⋅ 𝑥 6 )4
1
1
1
= (24 )4 ⋅ (32 )4 ⋅ (𝑥 6 )4
1
3
1
= 2 ⋅ 32 ⋅ 𝑥 2 = 2x ⋅ (3𝑥)2 = 2x√3𝑥
Exercises
Simplify each expression.
4
5
1. 𝑥 ⋅ 𝑥
6
5
2
6 5
5
4. (𝑚 )
7.
𝑝
1
𝑝3
3
2 4
3
4
2. (𝑦 )
3
8
5. 𝑥 ⋅ 𝑥
𝑥2
6
9. √128
1
𝑥3
3
14. √16
Chapter 6
6. (𝑠 )
1
8.
11. √288
13. √25 ⋅ √125
4
1 3
6
4
3
4
10. √49
7
3. 𝑝5 ⋅ 𝑝10
5
12. √32 ⋅ 3√16
3
6
15.
39
𝑎 √𝑏 4
√𝑎𝑏 3
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
6-6 Skills Practice
Rational Exponents
Write each expression in radical form, or write each radical in exponential form.
1
1
1. 36
2. 85
3. √51
4. √153
4
2
3
5. 123
6. √37
3
3
7. (𝑐 3 )5
8. √6𝑥𝑦 2
Evaluate each expression.
1
1
9. 325
10. 814
1
1
11. 273
12. 42
3
4
14. (−243)5
13. 162
1
3
15. 27 ⋅ 27
5
3
3
16.
4 2
( )
9
Simplify each expression.
12
3
2
17. 𝑐 5 ⋅ 𝑐 5
1
19. (𝑞2 )
21. 𝑥
6
11
3
⋅𝑥
1
23.
1
𝑦4
12
25. √64
Chapter 6
1
20. 𝑝5 ⋅ 𝑝 2
4
11
2
22.
𝑥3
1
𝑥4
1
𝑦2
16
18. 𝑚 9 ⋅ 𝑚 9
1
24.
𝑛3
1
1
𝑛6 ⋅ 𝑛2
8
26. √49𝑎 8 𝑏2
39
Glencoe Algebra 2