Download presentation

Presentation is loading. Please wait.

Published byArabella O’Neal’ Modified over 9 years ago

1
N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws

2
T ODAY ’ S O BJECTIVES Students will be able to demonstrate an understanding of powers with integral and rational exponents, including: 1. Explain, using patterns, why x -n = 1/x n, x ≠ 0 2. Apply the exponent laws 3. Identify and correct errors in a simplification of an expression that involves powers

3
R ECIPROCALS Any two numbers that have a product of 1 are called reciprocals 4 x ¼ = 1 2/3 x 3/2 = 1 Using the exponent law: a m x a n = a m+n, we can see that this rule also applies to powers 5 -2 x 5 2 = 5 -2+2 = 5 0 = 1 Since the product of these two powers is 1, 5 -2 and 5 2 are reciprocals So, 5 -2 = 1/5 2, and 1/5 -2 = 5 2 5 -2 = 1/25

4
P OWERS WITH N EGATIVE E XPONENTS When x is any non-zero number and n is a rational number, x -n is the reciprocal of x n That is, x -n = 1/x n and 1/x -n = x n, x ≠ 0 This is one of the exponent laws:

5
E XAMPLE 1: E VALUATING P OWERS WITH N EGATIVE I NTEGER E XPONENTS Evaluate each power: 3 -2 3 -2 = 1/3 2 1/9 (-3/4) -3 (-3/4) -3 = (-4/3) 3 -64/27 We can apply this law to evaluate powers with negative rational exponents as well Look at this example: 8 -2/3 The negative sign represents the reciprocal, the 2 represents the power, and the 3 represents the root

6
E XAMPLE 2: E VALUATING P OWERS WITH N EGATIVE R ATIONAL E XPONENTS Remember from last class that we can write a rational exponent as a product of two or more numbers The exponent -2/3 can be written as (-1)(1/3)(2) Evaluate the power: 8 -2/3 8 -2/3 = 1/8 2/3 = 1/( 3 √8) 2 1/2 2 1/4 Your turn : Evaluate (9/16) -3/2 (16/9) 3/2 = ( √16/9) 3 = (4/3) 3 = 64/27

7
E XPONENT L AWS Product of Powers a m x a n = a m+n Quotient of Powers a m /a n = a m-n, a ≠ 0 Power of a Power (a m ) n = a mn Power of a Product (ab) m = a m b m Power of a Quotient (a/b) m = a m /b m, b ≠ 0

8
A PPLYING THE E XPONENT L AWS We can use the exponent laws to simplify expressions that contain rational number bases When writing a simplified power, you should always write your final answer with a positive exponent Example 3: Simplifying Numerical Expressions with Rational Number Bases Simplify by writing as a single power: [(-3/2) -4 ] 2 x [(-3/2) 2 ] 3 First, use the power of a power law: For each power, multiply the exponents (-3/2) (-4)(2) x (-3/2) (2)(3) = (-3/2) -8 x (-3/2) 6

9
E XAMPLE 3 Next, use the product of powers law (-3/2) -8+6 = (-3/2) -2 Finally, write with a positive exponent (-3/2) -2 = (-2/3) 2 Your turn : Simplify (1.4 3 )(1.4 4 )/1.4 -2 1.4 3+4 /1.4 -2 = 1.4 7 /1.4 -2 = 1.4 7-(-2) = 1.4 9 We will also be simplifying algebraic expressions with integer and rational exponents

10
E XAMPLE 4 Simplify the expression 4a -2 b 2/3 /2a 2 b 1/3 First use the quotient of powers law 4/2 x a -2 /a 2 x b 2/3 /b 1/3 = 2 x a (-2)-2 x b 2/3-1/3 2a -4 b 1/3 Then write with a positive exponent 2b 1/3 /a 4 Your turn: Simplify (100a/25a 5 b -1/2 ) 1/2 (100/25 x a 1 /a 5 x 1/b -1/2 ) 1/2 (4a 1-5 b 1/2 ) 1/2 = (4a -4 b 1/2 ) 1/2 4 1/2 a (-4)(1/2) b (1/2)(1/2) = 2a -2 b 1/4 2b 1/4 /a 2

11
R EVIEW

12
R OOTS AND P OWERS H OMEWORK Page 227-228 #3,5,7,9,11,15,17-21 Extra Practice: Chapter Review, pg. 246 – 249 Review: Chapter 1-4, pg. 252 – 253 Finish Chapter 4 Vocabulary Book

Similar presentations

© 2024 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google