1. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 4 Polynomials

2. 1-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Negative Exponents and Scientific Notation Negative Integers as Exponents Scientific Notation Multiplying and Dividing Using Scientific Notation 4.2

3. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Negative Exponents For any real number a that is nonzero and any integer n, (The numbers a -n and a n are reciprocals of each other.)

4. 1-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Express using positive exponents, and, if possible, simplify. a) m  5 b) 5  2 c) (  4)  2 d) xy  1 Solution a) m  5 = b) 5  2 =

5. 1-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example c) (  4)  2 = d) xy  1 =

6. 1-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify. Do not use negative exponents in the answer. a) b) (x  4 )  3 c) (3a 2 b  4 ) 3 d)e) f) Solution a) b) (x  4 )  3 = x (  4)(  3) = x 12

7. 1-7 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example c) (3a 2 b  4 ) 3 = 3 3 (a 2 ) 3 (b  4 ) 3 = 27 a 6 b  12 = d) e) f)

8. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Factors and Negative Exponents For any nonzero real numbers a and b and any integers m and n, (A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.)

9. 1-9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify. Solution We can move the negative factors to the other side of the fraction bar if we change the sign of each exponent.

10. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Reciprocals and Negative Exponents For any nonzero real numbers a and b and any integer n, (Any base to a power is equal to the reciprocal of the base raised to the opposite power.)

11. 1-11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: Solution

12. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Scientific Notation Scientific notation for a number is an expression of the type N × 10 m, where N is at least 1 but less than 10 (that is, 1 ≤ N < 10), N is expressed in decimal notation, and m is an integer. Note that when m is positive the decimal point moves right m places in decimal notation. When m is negative, the decimal point moves left |m| places.

13. 1-13 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Convert to decimal notation: a) 3.842  10 6 b) 5.3  10  7 Solution a) Since the exponent is positive, the decimal point moves right 6 places. 3.842000. 3.842  10 6 = 3,842,000 b) Since the exponent is negative, the decimal point moves left 7 places. 0.0000005.3 5.3  10  7 = 0.00000053

14. 1-14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Write in scientific notation: a) 94,000b) 0.0423 Solution a) We need to find m such that 94,000 = 9.4  10 m. This requires moving the decimal point 4 places to the right. 94,000 = 9.4  10 4 b) To change 4.23 to 0.0423 we move the decimal point 2 places to the left. 0.0423 = 4.23  10  2

15. 1-15 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Multiplying and Dividing Using Scientific Notation Products and quotients of numbers written in scientific notation are found using the rules for exponents.

16. 1-16 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: (1.7  10 8 )(2.2  10  5 ) Solution (1.7  10 8 )(2.2  10  5 ) = 1.7  2.2  10 8  10  5 = 3.74  10 8 +(  5) = 3.74  10 3

17. 1-17 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify. (6.2  10  9 )  (8.0  10 8 ) Solution (6.2  10  9 )  (8.0  10 8 ) =