Learn how to use the multiplication properties of exponents to evaluate powers and simplify expressions

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1 8.1 Multiplication Properties of Exponents Objectives: Learn how to use the multiplication properties of exponents to evaluate powers and simplif expressions Learn how to use powers and the exponential change equation as models in real life settings a n = a a a a a n times a n a is the base; n is the power (n th power) KNOW! Multiplication rules when ou have the: 1) same base Product of Powers Propert a m a n = a m+n same base, add exponents ) powers of powers with same base Power of Power Propert (a m ) n = a mn multipl exponents ) power of product different bases Power of a Product Propert (a b) m = a m b m different bases, distribute power to each base and multipl Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

2 8.1 Multiplication Properties of Exponents Example #1: = + ( ) ( ) = 6 Example #: (x ) = x (x ) (x ) (x ) (x ) = x 8 Example #: (a) = a = 9a Exponential Equation: an equation with an exponent a x : When a variable a is the base and another variable x is the exponent, a is a change factor and x is the number of times the change occurs. When a > 1 the equation is a model for exponential growth x Growth When a < 1 the equation is a model for exponential deca x Deca Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

3 8.1 Multiplication Properties of Exponents Volume of a pramid is V = 1 Bh Where B is the area of the base: B = s Pramid of Cheops was 81 feet high and the square base was 75 feet on each side. How man cu. Ft. of space was there? V = 1 Bh = 1 (75) 81 = 91,000,000 ft Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

4 8. Negative and Zero Exponents Objectives: Learn to use negative and zero exponents in algebraic expressions Learn to use powers in real-life settings a n is the reciprocal of a n ( 1 ) ( ) a = or a Example: = 1 = = = n a n = 1 = n n a = The negative exponent sas the number needs to be moved to the opposite location (either numerator or denominator). If it s negative in the numerator, it belongs in the denominator position. If it s negative in the denominator position, it belongs in the numerator position. Numerator positive exponent stas put Denominator negative exponent move # Zero Exponent: an non-zero number to the zero power is ALWAYS 1 a 0 = 1 if a 0 0 = 1 ALWAYS! (x 5 ) 0 = 1 Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

5 8. Negative and Zero Exponents Example: Between 1970 and 1990 Missouri s population increased at a rate of 0.7% per ear. The population P in t ears is given b: P =,90, t 0.7% =.007 where t = 0 corresponds to 1980 Find the population in 1970, 1980 and 1990: P 1970 = =,678,06 P 1980 = =,90,000 P 1990 = = 5,18,76 Simplif expressions: write with positive exponents. ( 5) a b ( ) Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

6 8. Negative and Zero Exponents Graph = x x = x = x Notice that the line crosses at 1 and gets closer and closer to x as the x value gets smaller. It will never cross the x-axis x 1 Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

7 8. Division Properties of Exponents Objectives: Learn how to use division properties of exponents to evaluate powers and simplif expressions Learn how to use powers as models in real-life settings Reduce 5 = = 5 = Rules of Division: a and b an real number, and m and n be integers 1. To divide powers having the same base, subtract the exponents. a a m n = a m n, (a 0) Quotient of Powers Propert - subtract n from m. To find a power of a quotient, find the power of the numerator and the power of the denominator and divide. ( ) m m a = a m b b Examples:, (b 0) Power of Quotient Propert - distribute power across the ( ) Remember! a negative exponent means to re-write as its reciprocal! 1. 7 = 7 = = 7. ( ) = = 9. ( 1 ) 1 = = 1 x x x Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

8 8. Division Properties of Exponents. 88 = 8 = = = = 8 = 1 88 = 1 = = 1 6 = x 6x = x x 6 x = 6. 9x = 9 x = x = x 7. 6 x 8*. 5 6x 8 x x 6 8 8x = 1x = x = x 6 x 8 6 x x 1 x x x x x x 1 x x 9 x x = 9 6 * - Example #8 is the same as problem #8 on page 16 of the textbook Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

9 8. Scientific Notation Objectives: Learn how to use scientific notation to express large and small numbers Learn how to perform operations with numbers in scientific notation Scientific notation: used to write large and small numbers in powers of 10 Decimal Form Power of 10 Name 1,000,000, One billion 1,000, One million 1, One thousand Ten One One tenth One thousanth One millionth Scientific Notation Form: c.## 10 n where C is alwas a single-digit number between 1 and 9 1 c 9 Change decimals to scientific notation: 9,000, Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

10 8. Scientific Notation Scientific to decimal: ,000 (fill in the loops with zeros, count off for,) Multipling or Dividing scientific notation: answer must be in scientific notation form. Use exponent rules to combine powers of ten. ( )(. 10 ) (..) ( ) (1.) (10 10 ) ( ) (10 10 ) When dealing with fractions, change to decimals, then just move the decimal ( ) = 5 10 = 5 10,000 = Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

11 8.5 Problem Solving and Scientific Notation Objectives: Learn how to use scientific notation to solve real-life problems with ver large and ver small numbers Recommended Problem Solving Steps 1) write verbal model ) assign labels ) write algebraic model (plug in known factors) ) solve 5) answer the question Example #1: 1) population square km ) P = population = = people per square kilometer m = area in square km = x = people per square kilometer ) P m = x ) = = , people/km Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

12 8.5 Problem Solving and Scientific Notation Example #: 1) debt people = x amount/person ) P = Population = people (50,000,000 people) d = debt = dollars ($1,000,000,000, = $1 Trillion)) ) d P = x = = =. 10 = 10 = $,000 Example #: (see question #7 on page 9 of textbook) 1) D=rt t = D r ) D = distance of Jupiter from the sun = km r = speed of light = km/sec ) t = D r = = =.6 10 =,600 seconds = min. 0 sec. Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

13 8.5 Problem Solving and Scientific Notation Example #: (see question #8 on page 9 of textbook) The distance between Earth and the sun is kilometers. Write an inequalit that describes possible distances, d, between Earth and Jupiter km km Sun km km km km km km km km km d km Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

14 8.6 Problem Solving: Compound Interest Objectives: Learn how to use the compound interest formula Learn how to use models for exponential growth to solve real-life problems Simple interest: I = prt I = interest made p = principal: amount ou start with r = rate of interest t = time in ears Example: If ou invest $,000 at 5% for one ear, how much will ou make for the ear? I = prt = = 150 You will earn $150 for the ear. Compound Interest Formula: A = p(1 + r) t A = balance p = principal r = rate t = time in ears Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

15 8.6 Problem Solving: Compound Interest Example: Find the total amount in our account if ou start with $750 at 7.5% interest for.5 ears. A = p(1 + r) t = 750( ).5 = 750(1.075).5 (ou ma use a calculator!) = $898.6 Example: How much should ou invest at 7% to have $00 after 5 ears? A = p(1 + r) t (substitute what ou know, and solve for the unknown) 00 = p(1.07) 5 (ou ma use a calculator!) p = = $1.60 If ou put$100 in the bank at % interest and leave it alone for 0 ears, how much mone will ou have? What about a Bond that pas 10% interest? at %: 00 $,55.9 A = 100(1.0) 0 = $80.10 at 10%: A = 100(1.10) = $, $ x Mr. Noes, Akimel A-al Middle School Heath Algebra 1 - An Integrated Approach

16 8.7 Exponential Growth and Deca Objectives: Learn how to use models for exponential growth and deca to solve real-life problems growth = C(1 + r) t r > 1 deca = C(1 r) t r < 1 You bu a new 10 speed bike for $ It loses value at a rate of 15% per ear. What is it worth in ears? Worth = cost (rate) ears W = c(1 r) t W = 150(1.15) W = W 9.1 Less means 15% for total of.85 The population in Phoenix went from.5 million in 1980 to.5 million in What was its rate of growth. We need to solve for x.5 =.5(1 + x) 10.5 = (1 + x) ( 1 ) 10 = (1 + x) 1.8 ( ) = 1 + x x = % Mr. Noes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

Placement Test. Circle the letter of the correct answer.

Placement Test. Circle the letter of the correct answer. Week Unit 0. What is the place value where the numbers differ?,7,7 A ones tens hundreds thousands. 70 A 6,0 6,0,0,0. 7 A 00 909 999 99. Which of the following is equivalent to the product of? A ( ) (0