This document discusses rules for multiplying and dividing expressions with exponents. It states that to multiply terms with the same base, add the exponents. To divide terms with the same nonzero base, subtract the exponents. If subtracting exponents results in zero, the expression equals one. Negative exponents can be written as reciprocals with positive exponents. Examples are provided to illustrate simplifying expressions using these rules.

1. Warm Ups -7 • c • c • c • y • y = Is 45 divisible by 3? When solving a problem with exponents and grouping symbols, which do you do first? What is a composite number? What strategy do we use to find prime factors of a number?

2. Section 4.7: Exponents and Multiplication The Exponential Jungle. October 16 th Notes Need your thinking caps today so stay with me.

3. Multiplying Exponents: You Figure It Out 7 2  7 3 = 5 4  5 2 = b 7  b 1 = Hint: Expand the exponents.

4. Multiplying Exponents To multiply numbers OR variables with the same base, add the exponents. Arithmetic: 2 3  2 4 = 2 3+4 = 2 7 2  2  2  2  2  2  2 = 2 7 Algebra: p m  p n = p m+n = p m+n

5. So SIMPLIFY! 3  3 3 = a 5  a  b 2 = x 2  x 3  y  y 4 = -2x 2  3x 5 = 4x 2  3x 4 =

6. Finding a Power of a Power (7 2 ) 3 = Rule for Multiplying Powers with the Same Base. Raise a power to a power by multiplying the exponents. (7 2 ) 3 = (7 2 )  (7 2 )  (7 2 ) = 7 2+2+2 = 7 6 OR! (7 2 ) 3 = 7 2  3 = 7 6 Remember your Order of Operations. Get stuff done inside the PARENTHESES first, then the EXPONENTS!

7. Simplify each expression. (2 4 ) 2 = (c 5 ) 4 = (m 3 ) 2 =

8. Section 4.8: Exponents and Division The Exponential Jungle. October 16 th Notes

9. Dividing Powers with the Same Base. 7 8 /7 3 = 11 6 /11 5 =

10. Dividing Powers with the Same Base. To divide numbers or variables with the same nonzero base , you subtract exponents. Arithmetic: 4 5 /4 2 = 4 5-2 = 4 3 Algebra: a m /a n = a m-n Try these : 12m 5 /3m = 200m 200 /100m 100 =

11. Simplifying Expressions with Integer Exponents What happens when you divide powers with the same base and get zero as the exponent? Like this…write with a fraction bar. 3 4 /3 4 = 3 4 /3 4 = 3 4-4 = 0 = 3 0 ZERO AS AN EXPONENT Rule : 3 0 = 1 a 0 = 1, when a  0

12. Try These: (-8) 2 /(-8) 2 = 6b 3 /18b 3 = 5x 0 = 5 2 x 6 /5x 6 =

13. But..what about NEGATIVES and EXPONENTS!?!? What if you divide and you get a negative exponent? Write in expanded form with fraction bar. 3 2 /3 4 = 3 2 /3 4 = 3 2-4 = -2 = 3 -2 3 2 /3 4 = 1/3 2

14. Negative Exponents Arithmetic: 3 -2 = 1/3 2 Algebra: a -n = 1/a n If you forget why these rules are true, always write the example in an expanded form. That will remind you why these rules are true.

15. Need to Re-Write an Expression? When simplifying an expression with a negative exponent, you can write it as a fraction with a positive exponent: 4 -2 = 1/4 2 = 1/16 You can also write an expression with a fraction so that there is no fraction bar: 1/x 2 = x -2

16. Example Problems: Simplify as much as you can… 5 6 /5 8 = 4 5 /4 7 = a 4 /a 6 = 3y 8 /9y 12 =

17. Write without a Fraction Bar: x 2 y 3 /x 3 y = b 3 /b 9 = m 3 n 2 /m 6 n 8 = dr 5 /d 5 r 3 =

18. Assignment #25 Page 200: 11-27 odd. Page 206: 17-39 odd. START YOUR HOMEWORK NOW … If there is time… I highly recommend that you keep your notes open while you do your homework. That way you can look at your notes to remind yourself what to do when. If all else fails, ORDER OF OPERATIONS !