1. Bell Work: Add: 4x + a + c a + b x

2. Answer: 4x + ax + ca + cb x(a + b) 2

3. Lesson 53: Power rule for exponents, conversions of volume

4. In some of the previous problem sets there have been problems whose solutions have required the use of one or more of the following definitions and theorems. Definition:x  x  x….x = x(m is a natural number) Product Rule:x  x = x(x ≠ 0) Definition:x = x Definition:x = 1/x(x ≠ 0) Definition:x = 1(x ≠ 0) Quotient Rule:x /x = x(x ≠ 0) m mn m + n 1 -n 0 n mn – m

5. To complete the list of rules for exponents, we will introduce the power rule for exponents, which is a logical extension of the first definition. By using this definition, we can show that x Is to be used as a factor three times by writing x as x  x  x or (x). 33

6. If we wish to show that x should be used as a factor three times, we could write x  x  x = (x ) Since x  x  x equals x, then (x ) must also equal x. 5 55553 55515 53

7. Thus, when an exponential expression is raised to a power, we simplify by multiplying the exponents. We call this rule the power rule for exponents.

8. Power Rule for Exponents: if m and n are real numbers and x ≠ 0, then (x ) = x mnmn

9. Practice: a) (a ) b) (x ) c) (m ) 52 -4 -2 4

10. Answer: a)a b)x c)m 10 8 -8

11. Practice: a) (p ) b) (x ) c) (m ) -7 3 5 k 2

12. Answer: a) p b) x c) m 49 15 2k

13. The use of the power rule for exponents in expressions such as those before usually gives little trouble, but the use of this rule to simplify expressions such as (2x y z) is more complicated. We know that the notations indicates that 2x y z is to be used as a factor three times because 3 is the exported of the whole expression. 5 2 3 5 2

14. (2x y z)(2x y z)(2x y z) Since the order of multiplication of factors of a product does not affect the value of the product, we will rearrange the order of multiplication to get 2  2  2  x  x  x  y  y  y  z  z  z 5 2 5 2 5 2 5 5 5 2 2 2

15. This can ultimately be written as 2 x y z Which equals 8x y z 3 15 6 3 15 6 3

16. To obtain the same result by using the power rule for exponents, we would have to multiply the exponent of each facto of the given term by 3 or raise each factor of the given term to the third power. (2x y z) = (2) (x ) (y ) (z ) = 8x y z 5 2 3 3 5 3 2 3 3 15 6 3

17. To raise a term that contains no indicated additions to a given power, the exponent indicating the power is multiplied by the exponent of every factor of the numerator and the denominator(if there is one) of the term.

18. Practice: Solve (2a b z ) -2 2 -10 -5

19. Answer: 2 a b z a b z 32 -5 10 -10 50 10 -10 50

20. Practice: Solve 2 x y z 0 -2 4 15

21. Answer: x y z -30 60 15

22. We will not use unit multipliers to convert volume measurements.

23. Example: Use three unit multipliers to convert 180 cubic feet to cubic inches.

24. Answer: 180 ft  ft  ft  12in  12in  12in 1 ft 1 ft 1 ft = 180(12)(12)(12)in = 311040 in 3 3

25. Example: Use six unit multipliers to convert 800 cubic centimeters to cubic feet.

26. Answer: 800 (2.54)(2.54)(2.54)(12)(12)(12) =.02825 feet 3

27. HW: Lesson 53 #1-30