1. Lesson 1 MULTIPLYING MONOMIALS

2. What are we going to do…  Multiply monomials.  Simplify expressions involving powers of monomials.

3. Monomial monomial  A monomial is a number, a variable, or a product of a number and one or more variables. not  An expression involving the division of variables is not a monomial. constants  Monomials that are real numbers are called constants.

4. Examples of Monomials 1. -5 2. x 3. abc 3 4. 5xy 2 7 Not a monomial:4cd 3 9ab Why?

5. Rule #1: Product of Powers  To multiply two powers that have the same base, add the exponents.  For any number x, x m (x n ) = x m+n.  x 12 ● x 5 = x 17

6. Example 1  (5x 6 )(x 3 ) = 5(x 6 x 3 ) = 5x 9

7. Example 2  (4ab 4 )(-5a 2 b 3 ) = (4)(-5)(aa 2 )(b 4 b 3 ) = -20a 3 b 7

8. Example 3…on your own!  (2ab 5 )(-a 2 b) = (4)(-1)(aa 2 )(b 5 b) = -4a 3 b 6

9. Rule #2: Power of a Power  To find the power of a power, multiply the exponents.  For any number a, (a m ) n = a mn.  (a 4 ) 3 = a 12

10. Example 4  [(a 2 ) 3 ] 2 = [a 6 ] 2 = a 12 OR…. = [a] 2*3*2 = a 12

11. Example 5  (x 2 ) 4 = x 2*4 = a 8

12. Example 6…on your own!  [(3 2 ) 3 ] 2 = [3 6 ] 2 = 3 12 = 531,441

13. Rule #3: Power of a Product  To find the power of a product, find the power of each factor and multiply.  For all numbers x and y, (xy) m = x m y m.  (-2x 2 y 3 ) 3 = (-2) 3 (x 2 ) 3 (y 3 ) 3 = -8x 6 y 9  Think of it like distributing the exponent!

14. Example 7  (3ab) 3 = (3) 3 (a) 3 (b) 3 = 27a 3 b 3

15. Example 8  (5x 2 yz 3 ) 2 = (5) 2 (x 2 ) 2 (y) 2 (z 3 ) 2 = 25x 4 y 2 z 6

16. Example 9…on your own!  (-2x 3 yz 4 ) 2 = (-2) 2 (x 3 ) 2 (y) 2 (z 4 ) 2 = 4x 6 y 2 z 8

17. Simplifying Monomial Expressions  To simplify an expression involving monomials, write an equivalent expression in which: 1. each base appears exactly once 2. there are no powers of powers 3. all fractions are in simplest form

18. Example 10  [(8g 3 h 4 ) 2 ] 2 (2gh 5 ) 4 = [(8) 2 (g 3 ) 2 (h 4 ) 2 ] 2 (2) 4 (g) 4 (h 5 ) 4 = [64g 6 h 8 ] 2 (16g 4 h 20 ) = (64) 2 (g 6 ) 2 (h 8 ) 2 (16g 4 h 20 ) = 4096g 12 h 16 (16g 4 h 20 ) = (4096)(16)(g 12 g 4 )(h 16 h 20 ) = 65,536g 16 h 36

19. Example 11 (ab 4 )(ab 2 ) = (aa)(b 4 b 2 ) = a 2 b 6

20. Example 12…on your own! (-4c 4 d 4 )(4cd) = (-4)(4)(c 4 c)(d 4 d) = -16c 5 d 5

21. Example 13 (5a 2 b 3 c 4 )(4a 2 b 4 c 3 ) = (5)(4)(a 2 a 2 )(b 3 b 4 )(c 4 c 3 ) = 20a 2 b 7 c 7

22. Example 14 (7pq 7 ) 2 = (7) 2 (p) 2 (q 7 ) 2 = 49p 2 q 14

23. Example 15…on your own! (5x 3 ) 2 = (5) 2 (x 3 ) 2 = 25x 6

24. Example 16 (4cd) 2 (-2d 2 ) 3 = (4) 2 (c) 2 (d) 2 (-2) 3 (d 2 ) 3 = 16c 2 d 2 (-8d 6 ) = (16)(-8)(c 2 )(d 2 d 6 ) = -128c 2 d 8

25. Example 17 (2ag 2 ) 4 (3a 2 g 3 ) 2 = (2) 4 (a) 4 (g 2 ) 4 (3) 2 (a 2 ) 2 (g 3 ) 2 = 16a 4 g 8 (9a 4 g 6 ) = (16)(9)(a 4 a 4 )(g 8 g 6 ) = 144a 8 g 14

26. Review 1. When multiplying powers with the same base, we ______ the exponents. 2. When raising a power to a power, we _____________ the exponents. add multiply