Lesson 9 – 1 Rational Exponents Pre-calculus

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

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

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

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

Radical Form Two Truths & A Lie 2. Express in radical form & evaluate = ( 4 16 ) 3 = (2) 3 =8 B. 100 − 1 2 =− 1 10 = 1 100 1 2 = 1 100 = 1 10 C. (343) − 2 3 = 1 49 = 1 (343) 2 3 = 1 ( 3 343 ) 2 = 1 49

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

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

Radical Form Two Truths & A Lie 4. Evaluate A. 8 120 6 ≈36.26 B. 5 132 7 ≈930.69 C. 3 (−225) 2 ≈−36.99 =36.99 Square of a # is (+)!

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

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 4 7 𝑦 1 5 5 𝑥 3 14 𝑦 2 5 =10 𝑥 4 7 + 3 14 𝑦 1 5 + 2 5 =10 𝑥 11 14 𝑦 3 5

Radical Form Two Truths & A Lie 6. Simplify = 𝑥 5 2 − 𝑥 7 2 1 3 = ( 𝑥 2 ( 𝑥 1 2 − 𝑥 3 2 )) 1 3 Add exponents! B. 2 𝑥 3 4 𝑦 − 1 2 8 = 2 𝑥 6 𝑦 4 Mult exponents! = 256 𝑥 6 𝑦 4 = 2 8 𝑥 6 𝑦 −4 C. 3 4 𝑥 36 𝑦 −5 = 𝑥 3 𝑦 5 12 = 𝑥 36 𝑦 −5 1 4 1 3 = 𝑥 9 𝑦 − 5 4 1 3 = 𝑥 3 𝑦 − 5 12 = 𝑥 3 𝑦 5 12

Formulas in real world situations often are rational expressions. Radical Form 7. 𝑇=2𝜋 𝐿 1 2 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  0.995 m 𝑇=2𝜋 (0.995) 1 2 (9.8) − 1 2 ≈2.0 𝑠𝑒𝑐

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)

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