8.1 Apply Exponent Properties Involving Products. Learning Outcome To use properties of exponents involving products

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1 8.1 Apply Exponent Properties Involving Products Learning Outcome To use properties of exponents involving products Product of Powers Property Let a be a real number, and let m and n be positive integers. Words: To multiply powers having the same base, add the exponents Algebra: a m a n = a m + n Example: = = 5 9 Use the product of powers to simplify the expression a = b. w 9 w 2 w 7 = c = d. x 3 x 5 x 9 = e. x 5 x 4 y 8 y 7 = f j 7 j 6 j 4 = Power of a Power Property Let a be a real number, and let m and n be positive integers. Words: To find the power of a power, multiply exponents Algebra: (a m ) n = a m n Example: (5 6 ) 3 = = 5 18

2 Use the power of power property to simplify the expression a. (2 2 ) 3 = b. (w 9 ) 7 = c. (4 5 ) 8 = d. (x 3 ) 5 (x 9 ) 2 = e. [(x 5 ) 4 (y 8 ) 7 ] 2 = f. (7 2 ) 3 (j 7 ) 4 = Power of a Product Property Let a be a real number, and let m and n be positive integers. Words: To find the power of a product, find the power of each factor and multiply exponents Algebra: (ab) n = a n b n Example: (5 4) 3 = Use the product of powers to simplify the expression a. (2 5 5) 3 = b. (w 9 j 2 ) 7 = c. (8 5 4) 8 =

3 d. (3x 3 2y x) 3 = e. (5x 5 y 4 2xy 8 ) 3 = f. (3 2 3x 3 2xy) 4 = 8.2 Apply Exponent Properties Involving Quotients Learning Outcome To use properties of exponents involving quotients Quotient of Powers Property Let a be a nonzero real number, and let m and n be positive integers such that m > n To divide powers having the same base, subtract the exponents Algebra: a m = a m - n a n Example: 5 6 = = Use the quotient of powers property to simplify the expression a b. w 9 x 6 w 7 x 4

4 c. 4 5 a a 8 d. x 3 y 8 z 9 x 8 y 5 z 2 e. 5 2 x x 4 f. 7 2 j j 5 Power of Quotient Property Let a and b be real numbers with b 0 and let m be a positive integers To find a power of a quotient, find the power of the numerator and the power of the denominator and divide Algebra: ( a) m a m b Example: (5) 6 = b m w 9 x 6 w 7 x a a 8 2 x 3 y 8 z 9 x 8 y 5 z x x j j 5 7

5 8.3 Define and Use Zero and Negative Exponents Learning Outcome To use zero and negative exponents Definition of zero and negative exponents Evaluate the expression 2-3 (-10)

6 Properties of Exponents Evaluate the expression

7 8.4 Use Scientific Notation Learning Outcome To read and write numbers in scientific notation Scientific Notation A number is written in scientific notation when it is of the form c 10 n where 1< c < 10 and n is an integer. Write in scientific notation 78, 500, 000, , 000, 000, 000, , 000, 000 Write in Standard form 2 x x x x 10 8

8 Challenge x x x x 10-5 Order from least to greatest.52, 520 x 10-3,.0520, 520 x 10 3,.0052, 52 x 10 5 Evaluate each expression- write your answer in scientific notation ( )(l ) ( ) Write and Graph Exponential Growth Functions

9 Learning Outcome To write and graph exponential growth models Exponential function A function of the form y= ab x where a 0, b> 0, and b 1 Exponential growth A quantity that increases by the same percent over equal time periods Compound interest Interest earned on both an initial investment and on previously earned Write a rule for the function, then graph y= ab x b= interval a= y-intercept, Rule pay close attention to the y-value find the y-intercept y = x

10 Write a rule for the function, then graph Rule pay close attention to the y-value find the y-intercept y = x Write a rule for the function, then graph Rule pay close attention to the y-value find the y-intercept y = x

11 Exponential Growth Model y = a( 1 + r) t y = a(l + r) t a is the initial amount r is the growth rate (1 + r is the growth factor) t is the time period Example y = a(l + r) t You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 10 years? Example y = a(l + r) t You deposit $200 in a savings account that earns 3% interest compounded yearly. Find the balance in the account after the 8 years. Example y = a(l + r) t A business had $10,000 profit in Then the profit increased by 8% each year for the next 10 years. a. Write a function that models the profit in dollars over time. y = x b. Use the function to predict the profit in Write and Graph Exponential Decay Functions

12 Learning Outcome To write and graph exponential decay models Exponential function A function of the form y= ab x where a > 0 and 0 < b < 1 Exponential growth A quantity that decreases by the same percent over equal time periods Write a rule for the function, then graph y= ab x b= interval a= y-intercept, Rule pay close attention to the y-value find the y-intercept y = x Write a rule for the function, then graph Rule pay close attention to the y-value find the y-intercept y = x

13 Write a rule for the function, then graph Rule pay close attention to the y-value find the y-intercept y = x Exponential Growth Model y = a( 1 - r) t y = a(l - r) t a is the initial amount r is the decay rate (1 - r is the decay factor) t is the time period (do the math!)

14 Example y = a(l - r) t The population of a city decreased from 1995 to 2003 by 1.5% annually. In 1995 there were about 357,000 people living in the city. Write a function that models the city s population since Then find the population in Example y = a(l - r) t A business had 4000 employees in Each year for the next 5 years, the number of employees decreased by 2%.Write a function that models the number of employees over time. Example y = a(l - r) t An indoor water park had a declining attendance from 2000 to The attendance in 2000 was 18,000. Each year for the next 5 years, the attendance decreased by 5.5%. a. Write a function that models the attendance since 2005 y = x b. What was the attendance in 2005?

8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions.

8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions. 2. Use powers to model real life problems. Multiplication Properties of Exponents