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## Chapter 7: Exponents and Exponential Functions

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**Exponents**• Exponent – the number of times the base is multiplied by itself • EX: 35**1) Product of Powers Property**• When you multiply powers with like bases, ADD the exponents. • EX: 56· 53**EX:**• Simplify the expression. Write your answer using exponents. • (-7)2(-7)8 • x2·x6·x**2) Power of a Power Property**• When you raise a power to a power, MULTIPLY exponents. • EX: (x)3**EX:**• Simplify the expression. Write your answer using exponents. • (42)7 • [(-2)4]5 • [(m + 1)6]3**3) Power of a Product Property**• When a product is raised to a power, raise each factor to the power. • EX: (9xy)2**EX:**• Simplify each expression. Write your answer using exponents. • (20·17)3**EX: Simplify each expression.**• (-4x)2 • -(4x)2 • (2x3)2· x4 • (-10x6)2 · x2 • (3x5)3(2x7)2**Order of Magnitude**• The order of magnitude of a quantity is the power of 10 that is closest to the actual value of the quantity. • An estimate. • EX: 91,000 ≈ 100,000 ≈ 105**EX:**• A box of staples contains 104 stables. How many stables do 102 boxes contain?**EX:**• There are about 1 billion grains of sand in 1 cubic foot of sand. Use order of magnitude to find about how many grains of sand are in 25 million cubic feet of sand.**1) Quotient of Powers Property**• When dividing powers with like bases, subtract the exponents. • EX:**EX:**• Simplify the expression. Write your answer using exponents.**2) Power of a Quotient Property**• When a quotient is raised to a power, raise both the numerator and the denominator to the power and cancel if possible. • EX: (3/2)7**EX:**• Simplify the expression. • (-7/x)2 • (x2/4y)2 • (- 5/y)3 • (2s/3t)3· (t5/16) • (- 5/4)4 • (3x2/3y3)2**EX:**• The order of magnitude of the brightness of the Milky Way is 1036 watts. The order of magnitude of the brightness of a gamma ray burster is 1045 watts. How many times brighter is the gamma ray burster than the Milky Way? • http://www.youtube.com/watch?v=P2ESs1rPO_A**Zero Power**• Anything raised to the zero power is ONE. • EX: 40 = 1 • WHY:**Negative Exponents**• When you have a negative exponent in the numerator: • Put it in the denominator and make it positive. • EX: 4-3 • When you have a negative exponent in the denominator: • Put it in the numerator and make it positive. • EX: • NOTE: Negative exponents represent very small numbers.**EX: Evaluate the expression.**• Write your answer using only positive exponents.**EX: Simplify the expression.**• Write your answer using only positive exponents.**EX:**• The mass of one peppercorn is about 10-2 gram. About how many peppercorns are in a box containing 1kilogram of peppercorns?**Exponential Functions**• An exponential function is a function in the form of: y=abx • EX: y = 2·3x • They are nonlinear functions. • They have graphs that are curved NOT straight lines.**To write a rule for a function table:**• 1) Decide what value each y-value is being multiplied by. • This is b. • 2) Find the value of y when x=0. • This is a. • 3) Fill in a and b into y=abx.**Exponential Growth**• When a quantity increases by the same percent over equal time intervals. • EX: Each year the value of an antique car increases by 50%. • Exponential growth is different from linear growth because linear growth increases by the same amount each time interval, not by the same percent.**Exponential Growth Model**• a is the initial amount • (1 + r) is the growth factor • r is the growth rate (% written as a decimal) • t is the time period**EX:**• The owner of an original copy of a 1938 comic book sold it at an auction in 2005. The owner bought the comic book for $55 in 1980. The value of the comic book increased at a rate of 2.8% per year. • A) Write a function that models the value of the comic book over time. • B) What was the approximate value of the comic book at the time of the auction in 2005? Round your answer to the nearest dollar.**Compound Interest**• Interest earned on both an initial investment and on previously earned interest. • EX: You put $125 in a savings account that earns 2% interest compounded yearly. What will the balance in your account be after 5 years?**Exponential Decay**• When a quantity decreases by the same percent over equal time periods. • EX: The number of acres of forests in the U.S. decreases by 0.5% each year.**Exponential Decay Model**• a is the initial amount • (1-r) is the decay factor • r is the decay rate • t is the time period**EX:**• A farmer bought a tractor in 1999 for $30,000. The value of the tractor has been decreasing at a rate of 18% per year. • Write a function that models the value of the tractor over time. • What was the approximate value of the tractor in 2005?