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?