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Section 8 – 1 Zero and Negative Exponents • Any nonzero number raised to the zero power equals 1 a0 = 1 • For every nonzero number a and integer n, a n 1 n a • Notice that a negative exponent does NOT make anything negative, it moves the base and the exponent to the other part of the fraction • Ex1. Write as a simple fraction 5-3 • Ex2. Write as a simple fraction with no negative exponents 5a 2b3c 1 • Ex3. Write with no negative exponents 1 x 4 • Ex4. Evaluate 5x2y-3z4 when x = 2, y = 6, and z = -3 (write answer as a simple fraction) Section 8 – 2 Scientific Notation • Scientific notation is used to write large and small numbers in a way that is easier to read • A number in scientific notation is written as the product of two factors in the form a x 10n, where n is an integer and 1< a < 10 • It is ok to use x for multiplication when writing numbers in scientific notation • The exponent indicates how many places the decimal point was moved and in which direction • Ex1. Are the following numbers in scientific notation? If no, why not? A) 54.2 x 10-6 B) 4.32 x 108 C) 9.296 x 1023 D) .045 x 1011 • Ex2. Write each of the following numbers in scientific notation A) 45,600,000,000 B) .00000253 C) 459.2 x 1012 • The way we typically write numbers is standard notation • Ex3. Write 5.82 x 108 in standard notation Section 8-3 Multiplication Properties of Exponents • You can multiply numbers that have the same base by adding their exponents: am · an = am+n • Simplify. Write without negative exponents • Ex1. x3 · x5 · x · x-2 • Ex2. 4a4 · 5a-3 · a6 • Ex3. 2g-3 · 4h6 · g-2 · h-4 • To multiply numbers in scientific notation: 1) multiply the coefficients (the a) 2) multiply the powers of 10 by adding their exponents 3) convert to scientific notation • Ex4. Simplify. Write the answer in scientific notation. (5 x 106)(7 x 108) Section 8-4 More Multiplication Properties of Exponents • Raising a power to a power: For every nonzero number a and integers m and n, (am)n = amn • Simplify. Write without negative exponents. • Ex1. (x5)3 Ex2. a3(a5)-4 • Raising a product to a power: For every nonzero number a and b and integer n (ab)n = anbn • Notice with a product to a power, everything in the parentheses is raised to the exponent, whether it is a number or variable • Simplify. Write without negative exponents. • Ex3. (5x3y2)4 • Ex4. (3a2)-4(2a3b4)2 • Ex5. m-6(3m5)2 • Ex6. (4 x 105)3 Section 8 – 5 Division Properties of Exponents • When you divide powers with the same bases, subtract their exponents am mn a n a • Simplify. Write without negative exponents 5 12 x • Ex1. x Ex2. Ex3. a 2b3 12 5 x x a 4b5 • Ex4. (6.8 x 109) ÷ (4 x 106) • Raising a quotient to a power: For every n nonzero a and b and integer n, a a n b bn • Just like with products, everything in the parentheses is raised to the power • Simplify. Write without negative exponents 4 3 • Ex5. 3 Ex6. 3 4 8 • Ex7. 5x 3 y 4 Ex8. 2ab a3 2 5 Section 8 – 6 Geometric Sequences • A geometric sequence is one in which you can find consecutive numbers by multiplying by a common ratio • Remember that with arithmetic sequences you were adding a common difference, now you will be multiplying by a common number (called the common ratio) • If you cannot easily identify the common ratio, divide the 2nd number by the 1st and that is your common ratio • To test that it is a geometric sequence and that the ratio is constant, also divide the 3rd number by the 2nd (you should get the same number) • Find the common ratio • Ex1. 6, -18, 54, -162, … • Ex2. 64, 48, 36, 27, … • Ex3. Find the next two terms 8, 56, 392, 2744, … • The formula for a geometric sequence is A(n) = a · rn–1 • A(n) = nth term a = first term • r = common ratio n = term # • Ex4. Find the 5th and 12th terms A(n) = 4 · (-2)n–1 • Ex5. Determine whether the sequence is arithmetic or geometric A) 40, 20, 10, 5, … B) 40, 20, 0, -20, … • Read example 5 on page 426 Section 8 – 7 Exponential Functions • Exponential functions are in the form y = a·bx, where a is a nonzero constant, b is greater than 0 and b ≠ 1, and x is a real number • The graphs will be curves • Ex1. Evaluate y = 5 · 2x for x = -2, 2, 4 • Open your book to page 431 and look at Objective 2 Graphing Exponential Functions • When graphing, it is easiest to use a graphing calculator to create these tables for you • Ex2. Suppose 2 mice live in a barn. If the number of mice quadruples every 3 months, how many mice will be in the barn after 2 years? • Ex3. Graph. y = 2 · 3x Section 8 – 8 Exponential Growth and Decay • Exponential growth is y = a·bx with a > 0 and b >1 (a is still the starting amount and b is called the growth factor) • The growth factor must be greater than 1 for it to be exponential growth (this means that the amount is increasing) • Compound interest is a type of exponential growth (a = initial deposit, b = 100% + interest rate, x = # of interest periods) • Exponential decay uses the same model as exponential growth, except the growth factor b is between 0 and 1 (so it is called the decay factor) • The amount is decreasing in exponential decay (100% – rate of decrease) • Ex1. a) Suppose you deposit $1000 in a college fund that pays 7.2% interest compounded annually. Find the account balance after 5 years b) find the account balance if interest is paid quarterly instead of yearly • Ex2. Suppose the population of a certain endangered species has decreased 2.4% each year. Suppose there were 60 of these animals in a given area in 1999. a) Write an equation to model the number of animals in this species that remain alive in that area. b) Use your equation to find the approximate number of animals remaining in 2005.