1. 7.2 Properties of Rational Exponents
2/21/2014

2. Review of Fractions Adding Fractions with same denominators: 3 4 + 5 4
= 3+5 4
= 8 4
= 2

3. Find Least Common Multiple: 12
Review of Fractions
Adding Fractions with unlike denominators:
Multiples of 4: 4,8, 12
Multiples of 3: 3, 6, 9, 12
Find Least Common Multiple: 12
= 3∙ ∙4 12
= = 17 12

4. Review of Fractions
Multiplying fractions: you don’t need common denominators!
Multiply the numerators and denominators, then reduce if necessary.
3 4 × 2 3
= 3×2 4×3
= 6 12
= 1 2

6. Example 1 a. 62/3 61/3 = 6(2/3 + 1/3) = 63/3 = 61 = 6 • b. (33/4)4 =
Use Properties of Rational Exponents a.
62/3
61/3 =
6(2/3 + 1/3) =
63/3 = 61 = 6 • b.
(33/4)4 =
3(3/4 4) • = 33 = 27 c.
(16 •
25)1/2 =
161/2 •
251/2
= 16 ∙ 25 = 4 • 5 = 20 1 =
81/3 = = 2 1 d.
8 1/3 – e.
71/2
75/2 =
7(5/2 1/2) – =
74/2 = 72 = 49 6

7. Perfect Fourth 1 = 14 16 = 24 81 = 34 256 = 44 625 = 54 Fourth Root
4 1 =1
4 16 =2
4 81 =3
4 256 =4
4 625 =5

8. Perfect Fifth 1 = 15 32 = 25 243 = 35 1024 = 45 3125 = 55 Fifth Root
5 1 =1
5 32 =2
5 243 =3 =4 =5

9. Simplifying =5 =2 =3
In general

10. Properties of Radicals
Product property:
𝑛 𝑎∙𝑏 = 𝑛 𝑎 ∙ 𝑛 𝑏
Quotient property:
𝑛 𝑎 𝑏 = 𝑛 𝑎 𝑛 𝑏

11. Example 2 a. 3 • 9 = 3 • 9 = 3 • = 3 𝑜𝑟 3 3∙9 = 3 27 =𝟑 b. 48 3 = 3 48
Use Properties of Radicals a. 3 3 • 3 9 = 3 • 9
Product property of radicals = 3 •
Factor. = 3
Simplify.
𝑜𝑟 3 3∙9 = 3 27 =𝟑 b. 4 48 3 = 4 3 48
Quotient property of radicals = 2
Simplify. 11

12. Example 3
Write Radicals in Simplest Form a. 3 40 = 8 3 • 5
Factor out a perfect cube. =
Separate the product 8 3 • 5 = 2
Simplify. 5 3 b.
5 64
= 5 32∙2
Factor out a perfect fifth.
= 5 32 ∙ 5 2
=2 5 2 12

13. Simplify the expression.
Checkpoint
Use Properties of Radicals and Rational Exponents
Simplify the expression. 1. 2 3 • 4
ANSWER 2 2. 4 5 • 8
ANSWER 2 3. 54 3 2
ANSWER 3 4. 81 3
ANSWER 3

14. Example 5
Simplify Expressions with Variables
Simplify the expression. Write your answer using positive exponents only. Assume all variables are positive. a.
= 9 𝑥
= ∙ ( 𝑥 6 ) 1 2
9x 6
= 9 ∙ 𝑥 6∙ 1 2 =
3x 3 b.
4y 6 (
)1/2 =
41/2
y 6 (
)1/2 =
4 ∙ y (6 · 1/2) =
2y 3 14

15. = 3 𝑎 3 2 𝑐∙ 𝑐 2 𝑎 Example 5 = 𝑥 3 𝑦 6 1 3 = 𝑥 3∙ 1 3 𝑦 (6 ∙ 1 3 ) x 3
Simplify Expressions with Variables
= 𝑥 3 𝑦
= 𝑥 3∙ 𝑦 (6 ∙ 1 3 )
x 3 c. 3
y 6 = x
y 2
= 3 𝑎 𝑐∙ 𝑐 2 𝑎 d.
ac 2
3a 3/2 c –
3a (3/ )c 3 = –
=𝟑 𝒄 𝟑 𝒂
3a 1/2c 3 = or 15

16. =𝒚𝒛 𝒙𝒛 Example 5 𝑒. 𝑥 𝑦 2 𝑧 3 = 𝑥 𝑦 2 𝑧 3 1 2
𝑒. 𝑥 𝑦 2 𝑧 3
= 𝑥 𝑦 2 𝑧
= 𝑥 ∙ 𝑦 (2∙ ) ∙ 𝑧 (3 ∙ 1 2 )
= 𝒙 𝟏 𝟐 ∙𝒚 ∙ 𝒛 𝟑 𝟐
In radical form: any fractional exponent more than 1 (improper fraction), change fractional exponent to mixed number.
= 𝒙 𝟏 𝟐 ∙𝒚 ∙ 𝒛 𝟏 𝟏 𝟐
= 𝒙 𝟏 𝟐 ∙𝒚 ∙ 𝒛 𝟏 ∙ 𝒛 𝟏 𝟐 ∙
= 𝒚∙𝒛∙𝒙 𝟏 𝟐 ∙ 𝒛 𝟏 𝟐
=𝑦𝑧 𝑥 ∙ 𝑧
=𝒚𝒛 𝒙𝒛

17. Simplify the expression. Write your answer using
Checkpoint
Simplify Expressions with Variables
Simplify the expression. Write your answer using
positive exponents only. Assume all variables are
positive. 1.
25y 4
ANSWER
5y 2 2.
y 2
x 6
ANSWER y
x 3 3.
8u 3v 9 (
)1/3
ANSWER
2uv 3 4. 2x
x 1/3z 3 –
ANSWER
2x 2/3z 3

18. Homework: 7.2 p.362 #21-43 odd, 51-65 odd skip #35
“I didn’t like my beard at first. Then it grew on me!”