7-Chapter Notes Algebra 1

Multiplication properties of Exponents 7-1 Notes for Algebra 1 Multiplication properties of Exponents

7.1 pg. 395 21-63o, 69-84(x3)

Monomial (only one term) A number, a variable, or the product of a number and one or more variables with nonnegative integer exponents. An expression that involves division by a variable is not a monomial.

Constant A monomial that is a real number.

Example 1: Identify Monomials Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1.) 17−𝑐 2.) 8 𝑓 2 𝑔 3.) 3 4 4.) 5 𝑡

Example 1: Identify Monomials Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1.) 17−𝑐 No, it involves subtraction, so it has more than one term. 2.) 8 𝑓 2 𝑔 Yes, it involves the product of a number and two variables. 3.) 3 4 Yes, the expression is a constant. 4.) 5 𝑡 No, it has a variable in the denominator.

Product of Powers 𝑥 3 ∙ 𝑥 6 𝑥 3+6 𝑥 9 To multiply two powers that have the same base, add their exponents. 𝑥 3 ∙ 𝑥 6 𝑥 3+6 𝑥 9

Example 2: Product of Powers Simplify each expression. 1.) 𝑟 4 −12𝑟 7 2.) 6𝑐 𝑑 5 5 𝑐 5 𝑑 2

Example 2: Product of Powers Simplify each expression. 1.) 𝑟 4 −12𝑟 7 2.) 6𝑐 𝑑 5 5 𝑐 5 𝑑 2 −12 𝑟 11 30 𝑐 6 𝑑 7

Power of a Power To find the power of a power, multiply the exponents. 𝑥 3 6 𝑥 3∙6 𝑥 18

Example 3: Power of a Power Simplify 2 3 3 2

Example 3: Power of a Power Simplify 2 3 3 2 2 18

Power of a Product 2 𝑥 2 𝑦 5 𝑧 4 3 2 3 ∙ 𝑥 2∙3 ∙ 𝑦 5∙3 ∙ 𝑧 4∙3 To find the power of a product, find the power of each factor and multiply. 2 𝑥 2 𝑦 5 𝑧 4 3 2 3 ∙ 𝑥 2∙3 ∙ 𝑦 5∙3 ∙ 𝑧 4∙3 8𝑥 6 𝑦 15 𝑧 12

Example 4: Power of a Product Express the volume of a cube with side length 5𝑥𝑦𝑧 as a monomial.

Example 4: Power of a Product Express the volume of a cube with side length 5𝑥𝑦𝑧 as a monomial. 125𝑥 3 𝑦 3 𝑧 3

Simplify Expressions To simplify a monomial expression, write an equivalent expression in which: Each variables base appears exactly once, There are no powers of powers, and All fractions are in simplest form.

Example 5: Simplify Expressions

Example 5: Simplify Expressions

Division Property of Exponents 7.2 Notes for Algebra 1 Division Property of Exponents

7.2 pg. 403 19-57o, 66-87(x3)

Quotient of Powers 15𝑥 7 𝑦 3 45𝑥 3 𝑦 15 45 ∙ 𝑥 7−3 1 ∙ 𝑦 3−1 1 To divide two powers with the same base, subtract the exponents. 15𝑥 7 𝑦 3 45𝑥 3 𝑦 15 45 ∙ 𝑥 7−3 1 ∙ 𝑦 3−1 1 𝑥 4 𝑦 2 3

Example 1: Quotient of Powers Simplify. 𝑥 7 𝑦 12 𝑥 6 𝑦 3

Example 1: Quotient of Powers Simplify. 𝑥 7 𝑦 12 𝑥 6 𝑦 3 𝑥𝑦 9

Power of a Quotient 2 4 ∙ 𝑦 6∙4 3 4 ∙ 𝑥 3∙4 ∙ 𝑧 2∙4 16 𝑦 24 81𝑥 12 𝑧 8 To find the power of a quotient, find the power of a numerator and the power of the denominator. 2𝑦 6 3𝑥 3 𝑧 2 4 2 4 ∙ 𝑦 6∙4 3 4 ∙ 𝑥 3∙4 ∙ 𝑧 2∙4 16 𝑦 24 81𝑥 12 𝑧 8

Example 2: Power of a Quotient Simplify. 4𝑐 3 𝑑 2 5 3

Example 2: Power of a Quotient Simplify. 4𝑐 3 𝑑 2 5 3 64𝑐 9 𝑑 6 125

Zero exponent Any nonzero number raised to the zero power is equal to 1. 2𝑥 𝑦 3 𝑧 5 0 =1

Example 3: Zero Exponent Simplify. 1.) 12𝑚 8 𝑛 7 8𝑚 50 𝑛 10 0 2.) 𝑚 0 𝑛 3 𝑛 2

Example 3: Zero Exponent Simplify. 1.) 12𝑚 8 𝑛 7 8𝑚 50 𝑛 10 0 1 2.) 𝑚 0 𝑛 3 𝑛 2 𝑛

Negative Exponent Property For any nonzero number 𝑎 and any integer 𝑛, 𝑎 −𝑛 = 1 𝑎 𝑛 10 𝑥 −3 𝑦 6 25𝑥 7 𝑦 −6 𝑧 −2 10 25 ∙ 1 𝑥 7 𝑥 3 ∙ 𝑦 6 𝑦 6 1 ∙ 𝑧 2 1 2𝑦 12 𝑧 2 5𝑥 10

Example 4: Negative Exponents Simplify. 1.) 𝑥 −4 𝑦 9 𝑧 −6 2.) 75𝑝 3 𝑚 −5 15𝑝 5 𝑚 −4 𝑟 −8

Example 4: Negative Exponents Simplify. 1.) 𝑥 −4 𝑦 9 𝑧 −6 𝑦 9 𝑧 6 𝑥 4 2.) 75𝑝 3 𝑚 −5 15𝑝 5 𝑚 −4 𝑟 −8 5𝑟 8 𝑝 2 𝑚

Order of Magnitude (used to compare measures to estimate and preform rough calculations) The quantity of a number rounded to the nearest power of 10 970,000,000,000 10 12

Example 5: Apply Properties of Exponents Darin has $123,456 in his savings account. Tab has $156 in his savings account. Determine the order of magnitude of Darin’s account and Tab’s account. How many orders of magnitude as great is Darin’s account as Tab’s account?

Example 5: Apply Properties of Exponents Darin has $123,456 in his savings account. Tab has $156 in his savings account. Determine the order of magnitude of Darin’s account and Tab’s account. How many orders of magnitude as great is Darin’s account as Tab’s account? Darin’s: 10 5 , Tab’s: 10 2 , Darin’s account is 3 orders of magnitude as great as Tab’s account.

7-3 Notes for Algebra 1 Rational Exponents

7-3 pg. 410 17-83o, 96-114(x3)

Rational exponents For any nonnegative real number 𝑏, 𝑏 1 2 = 𝑏 17 1 2 17

Example 1: Radical and Exponential Forms Write each expression in radical form, or write each radical in exponential form. 1.) 81 1 2 2.) 38 3.) 12𝑚 1 2 4.) 32𝑤

Example 1: Radical and Exponential Forms Write each expression in radical form, or write each radical in exponential form. 1.) 81 1 2 2.) 38 3.) 12𝑚 1 2 4.) 32𝑤 9 38 1 2 12 𝑚 32𝑤 1 2

𝑛th root For any real numbers 𝑎 and 𝑏 and any positive integer 𝑛. If 𝑎 𝑛 =𝑏, then 𝑎 is the 𝑛th root of 𝑏. 6 64 2

Example 2: 𝑛th roots Simplify. 1.) 4 256 2.) 6 15,625

Example 2: 𝑛th roots Simplify. 1.) 4 256 4 2.) 6 15,625 5

𝑏 1 𝑛 For any positive real number 𝑏 and any integer 𝑛>1, 𝑏 1 𝑛 = 𝑛 𝑏 . 16 1 4 4 16 4 2∙2∙2∙2 2

Example 3: Evaluate 𝑏 1 𝑛 Expressions Simplify. 1.) 1331 1 3 2.) 2401 1 4

Example 3: Evaluate 𝑏 1 𝑛 Expressions Simplify. 1.) 1331 1 3 2.) 2401 1 4 11 7

𝑏 𝑚 𝑛 For any positive real number 𝑏 and any integer 𝑚 and 𝑛>1. 𝑥 3 7 7 𝑥 3 or 7 𝑥 3

Example 4: Evaluate 𝑏 𝑚 𝑛 Expressions Simplify. 1.) 32 2 5 2.) 81 5 2

Example 4: Evaluate 𝑏 𝑚 𝑛 Expressions Simplify. 1.) 32 2 5 2.) 81 5 2 4 59, 049

Power Property of Equality For any real number 𝑏>0 and 𝑏≠1, 𝑏 𝑥 = 𝑏 𝑦 if and only if 𝑥=𝑦. 7 𝑥 = 7 13 𝑥=13

Example 5: Solve Exponential Equations Solve each equation. 1.) 9 𝑥 =729 2.) 16 2𝑥−1 =8

Example 5: Solve Exponential Equations Solve each equation. 1.) 9 𝑥 =729 2.) 16 2𝑥−1 =8 9 𝑥 = 9 3 16 2𝑥−1 = 16 3 4 𝑥=3 2𝑥−1= 3 4 2𝑥= 7 4 𝑥= 7 8

Example 6: solve exponential equations BIOLOGY: The population 𝑝 of a culture that begins with 40 bacteria and doubles every 8 hours can be modeled by 𝑝=40 2 𝑡 8 , where 𝑡 is time in hours. Find 𝑡 if 𝑝=20,480.

Example 6: solve exponential equations BIOLOGY: The population 𝑝 of a culture that begins with 40 bacteria and doubles every 8 hours can be modeled by 𝑝=40 2 𝑡 8 , where 𝑡 is time in hours. Find 𝑡 if 𝑝=20,480. 20,480=40 2 𝑡 8 512= 2 𝑡 8 2 9 = 2 𝑡 8 9= 𝑡 8 72 ℎ𝑜𝑢𝑟𝑠

7-4 Notes for Algebra 1 Scientific Notation

7.4 pg. 414 21-61o, 78-93(x3)

Scientific Notation A way to calculate very large or small numbers easily.

Standard form to Scientific Notation Move the decimal point until it is to the right of the first nonzero digit. Note the number of places 𝑛 and the direction that you moved the decimal point. If the decimal point is moved left, write the number as 𝑎× 10 𝑛 . If the decimal point is moved right, write the number as 𝑎× 10 −𝑛 . Remove the unnecessary zeros.

Example 1: Standard Form to Scientific Notation Express each number in scientific notation. 1.) 4,062,000,000,000 2.) 0.000000823

Example 1: Standard Form to Scientific Notation Express each number in scientific notation. 1.) 4,062,000,000,000 2.) 0.000000823 4.062× 10 12 8.23× 10 −7

Scientific Notation to Standard Form In 𝑎× 10 𝑛 , note whether 𝑛>0 or 𝑛<0. If 𝑛>0, move the decimal point 𝑛 places right. If 𝑛<0, move the decimal point −𝑛 places left. Insert zeros, decimal point and commas as needed for place value.

Example 2: Scientific Notation to Standard Form Express each number in standard form. 1.) 6.49×1 0 5 2.) 1.8× 10 −3

Example 2: Scientific Notation to Standard Form Express each number in standard form. 1.) 6.49×1 0 5 2.) 1.8× 10 −3 649,000 0.0018

Example 3: Multiply with Scientific Notation Express the result in both Scientific Notation and standard form. Evaluate 5× 10 −6 2.3× 10 12

Example 3: Multiply with Scientific Notation Express the result in both Scientific Notation and standard form. Evaluate 5× 10 −6 2.3× 10 12 1.15×1 0 7 and 11,500,000

Example 4: Divide with Scientific Notation Express the result in both scientific notation and standard form. Evaluate 4.5×1 0 8 1.5× 10 10

Example 4: Divide with Scientific Notation Express the result in both scientific notation and standard form. Evaluate 4.5×1 0 8 1.5× 10 10 3× 10 −2 and 0.03

Example 5: Use Scientific Notation WATERCRAFT: Last year Alison’s state registered over 400 thousand watercraft. Boat sales in her state generated more than $15.4 million in state sales taxes that same year. Express the number of watercraft registered and the state sales tax generated from boat sales last year in Alison’s state in standard form. Write each number in Scientific Notation. How many watercraft have been registered in Alison’s state if 12 time the number registered in all? Write your answer in scientific notation and standard form.

Example 5: Use Scientific Notation WATERCRAFT: Last year Alison’s state registered over 400 thousand watercraft. Boat sales in her state generated more than $15.4 million in state sales taxes that same year. Express the number of watercraft registered and the state sales tax generated from boat sales last year in Alison’s state in standard form. Watercraft registered: 400,000 Sales tax generated: $15,400,000 Write each number in Scientific Notation. Water craft registered: 4× 10 5 Sales tax generated: 1.54× 10 7 How many watercraft have been registered in Alison’s state if 12 time the number registered in all? Write your answer in scientific notation and standard form. 4,800,000 4.8× 10 6

Exponential Functions 7-5 Notes for Algebra 1 Exponential Functions

7-5 pg. 427 11-33o, 48-66(x3)

Exponential Function A function that can be described by an equation of the form 𝑦=a 𝑏 𝑥 , where 𝑎≠0, 𝑏>0, and 𝑏≠1. Exponential Growth models 𝑦=3 2 𝑥 𝑦= 5 𝑥 Exponential Decay models 𝑦= 2 3 𝑥

Example 1: Graph with 𝑎>0 and 𝑏>1 Graph 𝑦= 4 𝑥 . Find the 𝑦-intercept State the domain and range.

Example 1: Graph with 𝑎>0 and 𝑏>1 Graph 𝑦= 4 𝑥 . Find the 𝑦-intercept State the domain and range. (0, 1) 𝐷= 𝐴𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅= 𝐴𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 7 6 5 4 3 2 1 -4 -3 -2 -1

Example 2: Graph with 𝑎>0 and 0<𝑏<1 Graph 𝑦= 1 4 𝑥 . Find the 𝑦-intercept State the domain and range.

Example 2: Graph with 𝑎>0 and 0<𝑏<1 Graph 𝑦= 1 4 𝑥 . Find the 𝑦-intercept State the domain and range. 0, 1 𝐷= 𝐴𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅= 𝐴𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 7 6 5 4 3 2 1 -4 -3 -2 -1

Example 3: Use Exponential Function to Solve Problem DEPRECIATION: Some people say that the value of a new car decreases as soon as it is driven off the lot. The function 𝑉=25,000∙ 0.82 𝑡 models the depreciation in the value of a new car that originally cost $25,000. 𝑉 represents the value of the car and 𝑡 represents the time in years from the time of purchase. Graph the function. What values of 𝑉 and 𝑡 are meaningful in the context of the problem? What is the cars value after five years?

Example 3: Use Exponential Function to Solve Problem 25000 20000 15000 10000 5000 1 2 3 4 5 6 7 8 9 Graph the function. What values of 𝑉 and 𝑡 are meaningful in the context of the problem? What is the cars value after five years? about $9270

Example 4: Identify Exponential Behavior Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. 𝑥 10 20 30 𝑦 25 62.5 156.25

Example 4: Identify Exponential Behavior Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. The domain values are at regular intervals, and the range values have a common factor of 2.5, so the set is probably exponential. Also, the graph shows rapidly increasing values of 𝑦 and 𝑥 increases. 𝑥 10 20 30 𝑦 25 62.5 156.25

7-6 Notes for Algebra 1 Growth and Decay

7-6 pg. 434 4-12

Equation for Exponential Growth 𝑦=a 1+𝑟 𝑡 𝑦– is the final amount 𝑎– is the initial amount 𝑟—is the rate of change expressed as a decimal (𝑟>0) 𝑡—is time

Example 1: Exponential Growth POPULATION: In 2008 the town of Flat Creek had a population of about 280,000 and a growth rate of 0.85% per year. Write an equation to represent the population of Flat Creek since 2008. According to the equation what will be the population of Flat Creek in the year 2025?

Example 1: Exponential Growth POPULATION: In 2008 the town of Flat Creek had a population of about 280,000 and a growth rate of 0.85% per year. Write an equation to represent the population of Flat Creek since 2008. 𝑦=280,000 1.0085 𝑡 According to the equation what will be the population of Flat Creek in the year 2025? about 304,731

Compound Interest Interest earned or paid on both the initial investment and previously earned interest. (applied exponential growth)

Equation for Compound Interest 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴– is the current amount 𝑃– is the (Principle) initial amount 𝑟—is the annual rate of interest expressed as a decimal (𝑟>0) 𝑛—is the number of times the interest is compounded each year 𝑡—is time

Example 2: Compound Interest COLLEGE When Jean was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. Jean will receive the money when she turns 18 to help with her college expenses. What amount of money will Jean receive from the investment.

Example 2: Compound Interest COLLEGE When Jean was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. Jean will receive the money when she turns 18 to help with her college expenses. What amount of money will Jean receive from the investment. She will receive about $3380

Equation for Exponential Decay 𝑦=a 1−𝑟 𝑡 𝑦– is the final amount 𝑎– is the initial amount 𝑟—is the rate of change expressed as a decimal (0<𝑟<1) 𝑡—is time

Example 3: Exponential Decay CHARITY During an economic recession, a charitable organization found that its donations dropped by 1.1% per year. Before the recession, its donations were$390,000. Write an equation to represent the charity’s donations since the beginning of the recession 𝐴=390,000 0.989 𝑡 Estimate the amount of the donations 5 years after the start of the recession. about $369,017

Geometric Sequences as Exponential Functions 7-7 Notes for Algebra 1 Geometric Sequences as Exponential Functions

7-7 pg. 441 14-29, 45-69(x3)

Geometric Sequence The first term is a nonzero and each term after the first is found by multiplying the previous term by a nonzero constant 𝑟 called the common ratio.

Example 1: Identify Geometric Sequences Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1.) 0, 8, 16, 24, 32, … 2.) 64, 48, 36, 27, …

Example 1: Identify Geometric Sequences Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1.) 0, 8, 16, 24, 32, … Arithmetic; the common difference is 8 2.) 64, 48, 36, 27, … Geometric; the common ratio is 3 4

Example 2: Find Terms of Geometric Sequences Find the next three terms in each geometric sequence. 1.) 1, −8, 64, −512, … 2.) 40, 20, 10, 5, …

Example 2: Find Terms of Geometric Sequences Find the next three terms in each geometric sequence. 1.) 1, −8, 64, −512, 4096, −32,768, 262,144 2.) 40, 20, 10, 5, 2.5, 1.25, 0.625 5 2 , 5 4 , 5 8

𝑛th term of a Geometric Sequence The 𝑛th term 𝑎 𝑛 of a geometric sequence with first term 𝑎 1 and common ratio 𝑟 is given by the following formula where 𝑛 is any positive integer and 𝑎 1 , 𝑟≠0. 𝑎 𝑛 = 𝑎 1 𝑟 𝑛−1

Example 3: Find the 𝑛th term of a Geometric Sequence 1.) Write an equation for the nth term of the geometric sequence 1, −2, 4, −8, … 2.) Find the 12th term of this sequence.

Example 3: Find the 𝑛th term of a Geometric Sequence 1.) Write an equation for the nth term of the geometric sequence 1, −2, 4, −8, … 𝑎 𝑛 =1 −2 𝑛−1 2.) Find the 12th term of this sequence. −2048

Example 4: Graph a Geometric Sequence ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour. Draw graph to represent how many pounds of the sculpture is left at each hour.

Example 4: Graph a Geometric Sequence ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour. Draw graph to represent how many pounds of the sculpture is left at each hour. 60 50 40 30 20 10 1 2 3 4 5 6 7 0, 50 1, 40 2, 32 4, 20.48 3, 25.6 6, 13.1072 5, 16.384 7, 10.48576

7-8 Notes for Algebra 1 Recursive Formulas

7-8 pg. 448 10-21

Recursive Formula Allows you to find the nth term of a sequence by performing operations to one or more of the preceding terms.

Example 1: Use a Recursive Formula Find the first five terms of the sequence in which 𝑎 1 =−8 and 𝑎 𝑛 =−2 𝑎 𝑛−1 +5, if 𝑛≥2.

Example 1: Use a Recursive Formula Find the first five terms of the sequence in which 𝑎 1 =−8 and 𝑎 𝑛 =−2 𝑎 𝑛−1 +5, if 𝑛≥2. 𝑎 2 =21 𝑎 2 =−2 −8 +5 𝑎 2 =−37 𝑎 2 =−2 21 +5 𝑎 2 =79 𝑎 2 =−2 −37 +5 𝑎 2 =−153 𝑎 2 =−2 79 +5

Writing Recursive Formulas Determine if the sequence is arithmetic or geometric by finding a common difference or common ratio. Write a recursive formula. Arithmetic Sequence 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Geometric Sequence 𝑎 𝑛 = 𝑟∙𝑎 𝑛−1 State the first term and domain for 𝑛

Example 2: Write Recursive Formulas Write a recursive formula for each sequence. 1.) 23, 29, 35, 41, … 2.) 7, −21, 63, −189, …

Example 2: Write Recursive Formulas Write a recursive formula for each sequence. 1.) 23, 29, 35, 41, … 𝑎 1 =23; 𝑎 𝑛 = 𝑎 𝑛−1 +6; 𝑛≥2 2.) 7, −21, 63, −189, … 𝑎 1 =7; 𝑎 𝑛 = −3∙𝑎 𝑛−1 ; 𝑛≥2

Example 3: Write Recursive and Explicit Formulas CARS The price of a car depreciates at the end of each year. a.) Write a recursive formula for the sequence b.) Write an explicit formula for the sequence Year Price ($) 1 12,000 2 7200 3 4320 4 2592

Example 3: Write Recursive and Explicit Formulas CARS The price of a car depreciates at the end of each year. a.) Write a recursive formula for the sequence 𝑎 1 =12,000; 𝑎 𝑛 =0.6 𝑎 𝑛−1 b.) Write an explicit formula for the sequence 𝑎 𝑛 =12,000 0.6 𝑛−1 Year Price ($) 1 12,000 2 7200 3 4320 4 2592

Example 4: Translate between Recursive and Explicit Formulas 1.) Write a recursive formula for 𝑎 𝑛 =2𝑛−4. 2.) Write an explicit formula for 𝑎 1 =84; 𝑎 𝑛 =1.5 𝑎 𝑛−1 ; 𝑛≥2

Example 4: Translate between Recursive and Explicit Formulas 1.) Write a recursive formula for 𝑎 𝑛 =2𝑛−4. 𝑎 1 =−2; 𝑎 𝑛 = 𝑎 𝑛−1 +2; 𝑛≥2 2.) Write an explicit formula for 𝑎 1 =84; 𝑎 𝑛 =1.5 𝑎 𝑛−1 ; 𝑛≥2 𝑎 𝑛 =84 1.5 𝑛−1 ; 𝑛≥2