Exponents and Exponential Functions CHAPTER 8

Introduction We will examine zero and negative exponents and evaluate exponential equations. We will learn about scientific notation. We will do problems involving scientific notation and other exponential expressions to illustrate multiplying and dividing powers, raising a power to a power, and raising products and quotients to a power. We will also examine geometric sequences. Finally, we will evaluate and graph exponential functions and apply this to modeling exponential growth and decay.

Zero and Negative Exponents (8.1) Two properties of exponents that will be important for our work with exponents. Here the base, a, cannot be zero. Zero as an Exponent For every nonzero number a, a 0 = 1.

Zero and Negative Exponents (8.1) Sample Problem Simplify a)4 -3 b)(-1.23) 0

Zero and Negative Exponents (8.1) An algebraic expression is written in its simplest form when it is written with only positive exponents. If the expression is a fraction in simplest form, the only common factor of the numerator and denominator is 1.

Zero and Negative Exponents (8.1) When evaluating an exponential expression, you can write the expression with positive exponents (that is simplify the expression) before substituting in the values. Sample Problem Evaluate 3m 2 t -2 for m = 2 and t = -3.

Scientific Notation (8.2) Scientific notation is a shorthand way that we can use to write very large or very small numbers. Use a positive exponent to write a number greater than 1. Use a negative exponent to write a number between 0 and 1. Examples a)3.4 x 10 6 b) 5.43 x c) 2.1 x

Scientific Notation (8.2) Sample Problem Is each number written in scientific notation? If not, explain. a) x b)0.84 x c)6.11 x 10 5

Scientific Notation (8.2) Sample Problem Write each number in scientific notation. a)56,900,000 b)

Scientific Notation (8.2) To convert a number from scientific notation back into standard notation we just reverse the process. For a positive exponent, move the decimal to the right however many spaces indicated by the exponent. For a negative exponent, move the decimal to the left however many spaces indicated by the exponent. Add in the place-holding zeros. Sample Problem Write each number in standard notation. a)1.55 x 10 6 b)2 x

Scientific Notation (8.2) We can use scientific notation to help us place numbers in order. Write each number in scientific notation. Order the powers of 10. Arrange the decimals with the same power of 10 in order. Write the original numbers in that order. Sample Problem Order x 10 7, 5.12 x 10 5, 53.2 x 10, and 534 from least to greatest.

Scientific Notation (8.2) We can multiply a number that is in scientific notation by another number. We will use the Associative Property of Multiplication (Section 1.8) in order to do this. If the product is less than 1 or greater than 10, rewrite the product in correct scientific notation form. Sample Problem Simplify. Write each answer using scientific notation. a)7(4 x 10 5 ) b)0.5(1.2 x )

Multiplication Properties of Exponents (8.3) We can write a power as a product of powers with the same base. Multiplying Powers with the Same Base For every non-zero number a and integers m and n, a m a n = a m+n Sample Problem Rewrite each expression using each base only once. a) = b) = c) =

Multiplication Properties of Exponents (8.3) We can also use the property of Multiplying Powers with the Same Base to simplify algebraic expression that involve exponents. Sample Problem Simplify each expression. a)x x 2 x 4 b)2n 5 3n -2

Multiplication Properties of Exponents (8.3) In section 8.2 we multiplied a number that was in scientific notation by a number that was not. We will now take a look at how to multiply two numbers that are both in scientific notation. We will use both the Commutative and Associative Properties of Multiplication along with the Property of Multiplying Powers with the same base to do so. Multiplying Powers with the Same Base a m a n = a m+n Commutative Property of Multiplication ab = ba Associative Property of Multiplication (ab)c = a(bc)

Multiplication Properties of Exponents (8.3) Sample Problem Simplify (7 x 10 2 )(4 x10 5 ). Write the answer in scientific notation.

More Multiplication Properties of Exponents (8.4) Raising a power to another power is the same as raising the base to the products of the exponents. Raising a Power to a Power For every non-zero number a and integers m and n, (a m ) n = a mn Sample Problem Simplify (x 3 ) 6

More Multiplication Properties of Exponents (8.4) When simplifying expressions with powers be sure to use the order of operations as well. Simplify expressions in parentheses that are being raised to a power before multiplying by expressions outside the parentheses. Sample Problem Simplify c 5 (c 3 ) -2

More Multiplication Properties of Exponents (8.4) This is the second property of exponent. We can use it in conjunction with other properties of algebra to help us simplify expressions with exponents in them. Raising a Product to a Power For every non-zero number a and b integer n, (ab) n = a n b n Sample Problem Simplify (2x 2 ) 4

More Multiplication Properties of Exponents (8.4) Sample Problem Simplify (x -2 ) 2 (3xy 2 ) 4

More Multiplication Properties of Exponents (8.4) We can use the property of raising a product to a power to solve problems involving scientific notation. Sample Problem All objects, even resting ones, contain energy. A raisin has a mass of 10-3 kg. The expression 10-3 (3 x 108)2 describes the amount of resting energy in joules the raisin contains. Simplify the expression.

Division Properties of Exponents (8.5)

We can use the property of Dividing Powers with the Same Base to divide numbers in scientific notation as well. This particular application is for real world situations as well.

Division Properties of Exponents (8.5)

Exponential Functions (8.7) Sample Problem Evaluate each exponential function. a)y = 5 x for {2,3,4} b)t(n) = 4 3 n {-3, 6}

Exponential Functions (8.7) To graph an exponential function: make a table of values plot the points connect the points to form a smooth curve

Exponential Functions (8.7) Sample Problem Graph y = 3 2 x XY

Exponential Growth and Decay (8.8) Exponential growth can be modeled with the function y = a b x for a > 0 and b >1 y = a b x starting amount (when x = 0) exponent The base, which is greater than 1, is the growth factor

Exponential Growth and Decay (8.8) Sample Problem Since 1985, the daily cost of patient care in community hospitals in the United States has increased about 8.1% per year. In 1985, such hospital costs were an average of $460 per day. What is the approximate cost per day in 2000?

Exponential Growth and Decay (8.8) Exponential decay can be modeled with the function y = a b x for a > 0 and 0 < b <1 y = a b x starting amount (when x = 0) exponent The base, which is between 0 and 1, is the decay factor

Exponential Functions (8.7) Sample Problem The half-life of a radioactive substance is the length of time it takes for one half of the substance to decay into another substance. To treat some forms of cancer, doctors use radioactive iodine. The half-life of iodine-131 is 8 days. A patient receives a 12 mCi (milllicuries, a measure of radiation) treatment. How much iodine-131 is left in the patient 16 days later?

Exponential Functions (8.7) Sample Problem Since 1980, the number of gallons of whole milk each person in the United States drinks each year has decreased 4.1% each year. In 1980, each person drank an average of 16.5 gallons of whole milk per year. a)Write an equation to model the gallons of whole milk drunk per person. b)Use your equation to find the approximate consumption per person of whole milk in 2000.

Exponents and Exponential Functions CHAPTER 8 THE END