1. Lesson 9 – 1 Rational Exponents
Pre-calculus

2. Learning Objective Express exponential expressions in radical form
Evaluate exponential & radical expressions
Simplify exponential expressions

3. Properties of Integer Exponents
Recall  Properties of Integer Exponents
Properties of Integer Exponents
𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛
𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 , 𝑏≠0
( 𝑎 𝑛 ) 𝑚 = 𝑎 𝑛𝑚
𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 , 𝑎≠0
(𝑎𝑏) 𝑛 = 𝑎 𝑛 𝑏 𝑛
𝑎 −𝑛 = 1 𝑎 𝑛 , 𝑎≠0
𝑎 0 =1, 𝑎≠0
Also recall  𝑛 𝑎
If n is odd  𝑛 𝑡ℎ root of 𝑎
3 −8 =−2
If n is even & 𝑎≥0
 nonnegative 𝑛 𝑡ℎ root of 𝑎
4 81 =3
If n is even & 𝑎<0
 Not a real number
−9 =3𝑖

4. Properties of Integer Exponents
𝑎 1 𝑛 and 𝑛 𝑎 represent the principal root of 𝑎
Properties of Integer Exponents
If m is an integer, n is a positive integer, and 𝑛 𝑎 is a real number, then
then 𝑎 1 𝑛 = 𝑛 𝑎 and 𝑎 𝑚 𝑛 = ( 𝑛 𝑎 ) 𝑚 = 𝑛 𝑎 𝑚

5. Throughout this lesson we are going to play “Two Truths and a Lie
Throughout this lesson we are going to play “Two Truths and a Lie.” It is based on the ice breaker game where you tell 3 things about yourself – 2 are true, 1 is not. We are doing it “Math Style” today! You will be asked to find a “lie” and fix it to make it a truth.
Radical Form
1. Express in radical form & evaluate (−125) 2 3
= ( 3 −125 ) 2
= (−5) 2
=25

6. Radical Form Two Truths & A Lie 2. Express in radical form & evaluate
= ( 4 16 ) 3
= (2) 3 =8
B − 1 2 =− 1 10 = =
= 1 10
C. (343) − 2 3 = 1 49
= 1 (343) 2 3
= 1 ( ) 2
= 1 49

7. Check – up
1. Express in radical form & evaluate
1 7 −128 4
=− 1 16

8. If you are asked to use your calculator & evaluate, be thoughtful about the parentheses needed. Get an error message? Maybe you need to augment how you enter it.
Radical Form
3. Evaluate using a calculator 5 −7
Two ways to enter it
5 𝑥 −7
(−7)^(1/5)
=−1.48

9. Radical Form Two Truths & A Lie 4. Evaluate A. 8 120 6 ≈36.26
B ≈930.69
C. 3 (−225) 2 ≈−36.99
=36.99
Square of a # is (+)!

10. Check – up
2. Evaluate using a calculator
3 −129
=−5.05

11. Properties of Exponents are also used in simplifying exponential expressions with variables in them.
Radical Form
Note: The order the problem is done can vary from person to person – Do what you are comfortable with!
5. Simplify 2 𝑥 𝑦 𝑥 𝑦 2 5
=10 𝑥 𝑦
=10 𝑥 𝑦 3 5

12. Radical Form Two Truths & A Lie 6. Simplify
= 𝑥 − 𝑥
= ( 𝑥 2 ( 𝑥 − 𝑥 )) 1 3
Add exponents!
B. 2 𝑥 𝑦 − = 2 𝑥 6 𝑦 4
Mult exponents!
= 256 𝑥 6 𝑦 4
= 2 8 𝑥 6 𝑦 −4
C 𝑥 36 𝑦 −5 = 𝑥 3 𝑦 5 12
= 𝑥 36 𝑦 −
= 𝑥 9 𝑦 −
= 𝑥 3 𝑦 − 5 12
= 𝑥 3 𝑦 5 12

13. Formulas in real world situations often are rational expressions.
Radical Form
7. 𝑇=2𝜋 𝐿 𝑔 − 1 2
𝑇= time in seconds to complete 1 pendulum swing
𝐿= length of pendulum
𝑔= acceleration due to gravity ≈9.8 𝑚/ 𝑠 2
Determine the period of a pendulum on a clock that is 99.5 cm long
99.5 cm  m
𝑇=2𝜋 (0.995) (9.8) − 1 2
≈2.0 𝑠𝑒𝑐

14. Check – up = 𝑥 3 (if n is even) and −𝑥 3 (if n is odd) 3. Simplify
𝑛 −𝑥 𝑛 3
= 𝑥 3 (if n is even)
and −𝑥 3 (if n is odd)

15. Assignment
9-1: Pg. 443 #1 – 53 odd (skip 17, 31)