6 7 6 y 7 8 0 y 7 8 0 Locker LESSON 1 1 Using Graphs and Properties to Solve Equations with Eponents Common Core Math Standards The student is epected to: A-CED1 Create equations and inequalities in one variable including ones with absolute value and use them to solve problems Include equations arising from linear and quadratic functions, and simple rational and eponential functions Also A-SSEc, A-REI11, F-BF1, F-LE Mathematical Practices MP Using Tools Language Objective Eplain to a partner how to use a graph to find the solution to an equation with a variable eponent ENGAGE Essential Question: How can you solve equations involving variable eponents? When the bases are equal, use the Equality of Bases Property When the bases are not equal, graph each side of the equation as its own function and find the intersection PAGE 9 Name Class Date 11 Using Graphs and Properties to Solve Equations with Eponents Essential Question: How can you solve equations involving variable eponents? Eplore 1 Solving Eponential Equations Graphically In previous lessons, variables have been raised to rational eponents and you have seen how to simplify and solve equations containing these epressions How do you solve an equation with a rational number raised to a variable? In certain cases, this is not a difficult task If it is easy to see that since In other cases, like (), where would you begin? Let s find out A Solve for () for B Let ƒ () () Complete the table for ƒ () D f () 1 6 1 8 6 19 7 8 Let g () Complete the table for g () g () 1 6 7 C E Using the table of values, graph ƒ () on the aes provided y (, ) - 7 8 (, 8) (, ) f() (1, 6) (, 1) 0 1 6 7 8 Using the table, graph g () on the same aes as ƒ () y g() (, ) 7 f() 8-0 1 6 7 8 Resource Locker PREVIEW: LESSON PERFORMANCE TASK View the Engage section online Discuss why a town government might need to know the rate at which the town s population is growing Then preview the Lesson Performance Task F The graphs intersect at point(s): (, ) This means that ƒ () g () when Module 1 709 Lesson 1 DO NOT EDIT--Changes must be made through File info CorrectionKeyNL-A;CA-A Name Class Date 1 1 Using Graphs and Properties to Solve Equations with Eponents Essential Question: How can you solve equations involving variable eponents? A-CED1 For the full tet of these standards, see the table starting on page CA Also A-SSEc, A-REI11, F-BF1, F-LE Eplore 1 Solving Eponential Equations Graphically In previous lessons, variables have been raised to rational eponents and you have seen how to simplify and solve equations containing these epressions How do you solve an equation with a rational number raised to a variable? In certain cases, this is not a difficult task If it is easy to see that since In other cases, like (), where would you begin? Let s find out Solve for () for Let ƒ () () Complete the table for ƒ () f () 1 8 19 8 1 6 Let g () Complete the table for g () g () 1 7 Using the table of values, graph ƒ () on the aes provided (, ) Resource (, 8) f() (, ) (1, 6) (, 1) - 1 6 7 8 Using the table, graph g () on the same aes as ƒ () (, ) f() 1 6 7 8 The graphs intersect at point(s): This means that ƒ () when - g() (, ) g () Module 1 709 Lesson 1 IN1_MNLESE8976_U6M1L1 709 19/0/1 10:8 AM HARDCOVER PAGES 9 66 Turn to these pages to find this lesson in the hardcover student edition 709 Lesson 1 1
Reflect 1 Discussion Consider the function h () - Where do ƒ () and h () intersect? The graphs would not intersect as f () is always greater than 0 Raising any positive number to a positive eponent yields a positive number Divide the equation ( ) by on both sides (an Algebraic Step) and utilize the same method as in Eplore 1 to graph each side of the equation as a function The point of intersection would be: (, ) Is this the same point of intersection? Is this the same answer? Can this be done? Elaborate as to why or why not It is not the same point of intersection The y-values of the points are different They do represent the same solution because the equations are equivalent by the Division Property of Equality Eplore Solving Eponential Equations Algebraically Recall the eample, with the solution What about a slightly more complicated equation? Can an equation like () 160 be solved using algebra? A Solve () 160 for The first step in isolating the term containing the variable on one side of the equation is to divide each side of the equation by ( ) 160 B Simplify () Reflect C Rewrite the right hand side as a power of () () D Solve Discussion The last step of the solution process seems to imply that if b b y then y Is this true for all values of b? Justify your answer No, it is not true For eample, 0 0 8 but 8, or 1 7 1 98 but 7 98 In Reflect, we started to solve () algebraically Finish solving for () PAGE 60 EXPLORE 1 Solving Eponential Equations Graphically INTEGRATE TECHNOLOGY Students can use graphing calculators to solve an eponential equation by the method shown in the Eplore activity Students should enter the appropriate eponential function and constant function, graph both functions, and use the calculator s intersect feature to find their point of intersection QUESTIONING STRATEGIES When solving an eponential equation of the form ab c graphically, what two functions do you graph? the eponential function f () ab and the horizontal line g () c How does graphing these two functions on the same grid help you determine the value of the eponent? The value of the eponent is the -value of the point of intersection EXPLORE Solving Eponential Equations Algebraically Module 1 710 Lesson 1 PROFESSIONAL DEVELOPMENT Learning Progressions In this lesson, students continue to build on their understanding of geometric sequences and eponential functions They learn the Equality of Bases Property, which states that If b > 0 and b 1, then b b y if and only if y They learn to solve equations involving variable eponents either by using the Equality of Bases Property or by graphing They also begin to model real-world situations using eponential equations, which can then be solved by either method Work with eponential functions will continue as students learn about eponential growth and decay models and eponential regression QUESTIONING STRATEGIES In the equation 6, how can you evaluate? Write 6 as a power of 6, so Assuming that is an integer in the equation b c, what must be true for this method to work? The value of c must be a power of b Using Graphs and Properties to Solve Equations with Eponents 710
EXPLAIN 1 Solving Equations by Equating Eponents QUESTIONING STRATEGIES In an equation such as 6 () 76, what property of equality can you use to isolate? Eplain how Division Property of Equality; divide both sides by 6 How does this step compare to isolating a variable on one side of a linear equation? It is done for the same reason By isolating the power, you have isolated the variable as well Then you can compare the eponents in the final equivalent epression PAGE 61 Eplain 1 Solving Equations by Equating Eponents Solving the previous eponential equation for used the idea that if, then This will be a powerful tool for solving eponential equations if it can be generalized to if b b y then y However, there are values for which this is clearly not true For eample, 0 7 0 but 7 If the values of b are restricted, we get the following property Equality of Bases Property Two powers with the same positive base other than 1 are equal if and only if the eponents are equal Algebraically, if b > 0 and, b 1, then b b y if and only if y Eample 1 Solve by equating eponents and using the Equality of Bases Property _ ( ) 0 _ _ () 0 _ ( _ ) 6 Simplify Multiply both sides by _ Rewrite the right side as a power of _ 0 7 ( _ ) _ 0 _ Equality of Bases Property 7_ Divide both sides by Focus on Reasoning MP Discuss with students the limitations on the Equality of Bases Property Have students give eamples to show why the property does not apply when the base is 0, 1, or 1 For eample: 0 8 0 0 but 8; 1 0 99 1 1 but 0 99; and ( 1) ( 1) 6 1 but 6 AVOID COMMON ERRORS Some students may misread the base b in an epression b as a coefficient of and try to divide both sides of the equation by b to isolate the variable Remind them that when a number is raised to a power, it cannot be treated as a single factor They must use the properties of equality to isolate b, then use the Equality of Bases Property to solve for the variable Reflect ( _ ) ( _ ) 1 _ 7 Simplify Rewrite the right side as a power of _ Equality of Bases Property Suppose while solving an equation algebraically you are confronted with: 1 ( _ ) Can you find using the method in the eamples above? No, you cannot It is not possible because 1 is not a whole number power of Module 1 711 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs Have each student write an equation involving a variable eponent in the form b c After students echange equations, each partner should first decide whether c can be epressed as a whole number power of b If so, the student should rewrite c as a power of b and solve for If not, the student should use a graphing calculator to graph each side of the equation as a separate function and use the intersect feature to find the -coordinate of the intersection point, which is the solution to the original equation Have students check each other s work 711 Lesson 1 1
Your Turn Solve by equating eponents and using the Equality of Bases Property 6 _ () 18 7 8_ ( ) () 18 ( _ ) 8 _ () 18 _ ( _ ) 7 Eplain ( _ ) 16 9 ( _ Solving a Real-World Eponential Equation by Graphing Some equations cannot be solved using the method in the previous eample because it isn t possible to write both sides of the equation as a whole number power of the same base Instead, you can consider the epressions on either side of the equation as the rules for two different functions You can then solve the original equation in one variable by graphing the two functions The solution is the input value for the point where the two graphs intersect ( _ ) ) 8_ PAGE 6 EXPLAIN Solving a Real-World Eponential Equation by Graphing QUESTIONING STRATEGIES If a population grows by % each year, by what factor is the population multiplied each year? Eplain 10; if the population is p one year, it will be p + 00p 10p the net year Why is it appropriate to round a prediction involving time to the nearest year? A prediction is usually just an estimate, so rounding is appropriate Eample Solve by graphing two functions An animal reserve has 0,000 elk The population is increasing at a rate of 8% per year There is concern that food will be scarce when the population has doubled How long will it take for the population to reach 0,000? Analyze Information Identify the important information The starting population is 0,000 The ending population is 0,000 The growth rate is 8% or 008 Formulate a Plan With the given situation and data there is enough information to write and solve an eponential model of the population as a function of time Write the eponential equation and then solve it using a graphing calculator Set ƒ () the target population and g () the eponential model Input Y 1 ƒ () and Y g () into a graphing calculator, graph the functions, and find their intersection Image Credits: James Prout/Alamy Module 1 71 Lesson 1 DIFFERENTIATE INSTRUCTION Graphic Organizers Have students complete a graphic organizer that shows when to solve an equation involving a variable eponent algebraically and when to solve it graphically Solving b c (where b > 0, b 1, and c > 0) Value of c Solution Method c b d for some Algebraic: whole number d b b d, so d c is not a whole number power of b Graph: intersection of ƒ () b and g () c Using Graphs and Properties to Solve Equations with Eponents 71
INTEGRATE TECHNOLOGY When solving eponential equations graphically, have a student demonstrate how to identify the two functions to be graphed, enter them into a graphing calculator, and find the solution by finding the point of intersection Discuss how to adjust the viewing window so that the graph and the point of intersection are clearly visible PAGE 6 Solve Write a function P (t) ab t, where P (t) is the population and t is the number of years since the population was initially measured a represents a b represents b 0,000 108 the initial population of elk the yearly growth rate of the elk population t 0,000 The function is P (t) ( 108 ) To find the time when the population is 0,000, set the function or P (t) equal to 0,000 and solve for t t 0,000 0,000 ( 108 ) Write functions for the epressions on either side of the equation ƒ () 0,000 g () 0,000 (108) Using a graphing calculator, set Y 1 ƒ () and Y g () View the graph Use the intersect feature on the CALC menu to find the intersection of the two graphs The approimate -value where the graphs intersect is 900668 Therefore, the population will double in just a little over 9 years Justify and Evaluate Check the solution by evaluating the function at t 9 P ( 9 ) 0,000 (108) 9 0,000 ( 19990 ) 9,980 Since 9,980 0,000, it is accurate to say the population will double in 9 years 9 This prediction is reasonable because 108 Module 1 71 Lesson 1 71 Lesson 1 1 LANGUAGE SUPPORT Connect Contet Support students in interpreting the language used in problem statements Eplain that the word suppose at the beginning of a problem signals that what follows is a hypothetical eample, meaning that readers should use their imaginations to consider a possible scenario Often, a problem will be followed by the question, Why or why not? Eplain that the question is phrased this way so as not to give away the answer Students should understand that they need to eplain either why a result is true or why it is not true, depending on the situation
Your Turn Solve using a graphing calculator 8 There are wolves in a state park The population is increasing at the rate of 1% per year You want to make a prediction for how long it will take the population to reach 00 Graph Y 1 00 Y (11) 9 There are 17 deer in a state park The population is increasing at the rate of 1% per year You want to make a prediction for how long it will take the population to reach 00 Graph Y 1 00 Y 17 (11) Elaborate The intersection point is (711, 00) The wolf population will reach 00 in approimately 7 years The intersection point is (7606, 00) The deer population will reach 00 in approimately 8 years 10 Eplain how you would solve 0 0 Which method can always be used to solve an eponential equation? Possible answer: Algebraically ELABORATE QUESTIONING STRATEGIES What is the shape of the graph of an eponential function of the form f () b when b > 1? It is a curve that rises in greater and greater amounts as increases What is the shape of the graph of a function ƒ () b when 0 < b < 1? It is a curve that falls more and more gradually as increases 0 0 (0) (0) Eponential equations can always be solved graphically 11 What would you do first to solve the equation 1 (6)? Multiply each side of the equation by to isolate the power 1 How does isolating the power in an eponential equation like 1 (6) compare to isolating the variable in a linear equation? Both are done for the same reason By isolating the power, you have isolated the variable as well Then you can compare the eponents in the final equivalent epressions 1 Given a population decreasing by 1% per year, when will the population double? What will this type of situation look like when graphed on a calculator? It will never double as the population is decreasing The equation representing this situation, 099, has no solution for > 0 The graphing calculator will show a horizontal line at and an eponential function with a y-intercept of 1 decreasing towards the positive -ais 1 Solve 0 101 graphically Suppose this equation models the point where a population increasing at a rate of 1% per year is halved When will the population be halved? Since -696607, you would have to go back in time, which is not possible Seventy years or so ago the population was half of what it is now SUMMARIZE THE LESSON How can you solve an equation where the variable is an eponent? First, use the properties of equality to isolate the number raised to a variable power Then check whether the constant on the other side of the equation can be written as a whole number power of the same base If it can, use the Equality of Bases Property to solve If not, graph each side of the equation as its own function and find the -value of the point of intersection 1 Essential Question Check-In How can you solve equations involving variable eponents? When the bases are equal, use the Equality of Bases Property When there are not equal bases on both sides of the equation, graph each side of the equation as its own function and find the intersection Module 1 71 Lesson 1 Using Graphs and Properties to Solve Equations with Eponents 71
EVALUATE PAGE 6 Evaluate: Homework and Practice 1 Would it have been easier to find the solution to the equation in Eplore 1, (), algebraically? Justify your answer In general, if you can solve an eponential equation graphing by hand, why can you solve it algebraically? Online Homework Hints and Help Etra Practice Yes, () becomes () after dividing both sides of the equation by and is an integer power of ASSIGNMENT GUIDE In general, the input-output tables for f () and g () have integers in the domain and the values in the range are easy to calculate Concepts and Skills Eplore 1 Solving Eponential Equations Graphically Eplore Solving Eponential Equations Algebraically Practice Eercises Eercises 1, The equation (101) models a population that has doubled What is the rate of increase? What does represent? The rate of increase is 1% per unit time is number of units of time Can we solve equations using both algebraic and graphical methods? Yes We can simplify the equation algebraically and then use graphing Eample 1 Solving Equations by Equating Eponents Eample Solving a Real-World Eponential Equation by Graphing Eercises 16, Eercises 17 Solve the given equation () 6 7 () 6 6 () 6 16 7 () 6 7 7 9 _ 6 6 6 16 6 6 Focus on Communication MP Circulate as students solve the practice eercises Invite students to eplain their reasoning as they begin a new problem 7 1 ( ) ( 6) ( 1 7 ) ( _ 6) ( _ 6) 6 ( _ 6) ( _ 6) _ 7 8 7_ ( ) _ 9 ( 7_ ) 9 ( 7_ ) 9 ( 7_ ) ( 7_ ) 9 (11) 99 (11) 99 11 11 11 11 Module 1 71 Lesson 1 Eercise Depth of Knowledge (DOK) Mathematical Practices 1 Skills/Concepts MP6 Precision Skills/Concepts MP Modeling Skills/Concepts MP6 Precision 1 1 Recall of Information MP Reasoning 1 16 Skills/Concepts MP Reasoning 17 Skills/Concepts MP Modeling 71 Lesson 1 1
10 (9) 16 11 1_ ( 9 81 9 9 1 1 ( ) ( ) ( 1 16 ) ( ( 1_ ) ( _ ) ( 1_ ) ( 16 ( _ ) 8 7 ( _ ) ( _ ) 16 _ ( ) ( ) _ ( _ ( _ ( _ ) ( 8 ) (( _ ) ) ( _ ) 6 6 6_ ( 8_ 8_ 1) ( 1) _ 1) 9) ( 1_ 9) _ 81 81 17 There is a drought and the oak tree population is decreasing at the rate of 7% per year If the population continues to decrease at the same rate, how long will it take for the population to be half of what it is? The model for the oak tree population is P (t) P i (09) t, where t is the time in years, P i is the initial population, and P (t) is the population in year t To find when the population is half of its initial value, solve P (t) P i for t ( 1_ ( 1_ 9) 1 81 9) ( 1_ 9) 1 _ ( _ 1) 169 ( 1) 169 ( 1) _ 16 169 ( 1) ( 1) 7) 1 (8) _ ( () ) ( _ 16 7) 1 _ ( 8 7) ( _ ) ( _ ( _ 16 7) 8 8 ( _ 7) ) ) ( 8 ) ( _ ) ( _ Using a calculator graph each side of the equation 1_ (09) t as a _ ) ( 8_ ) 1 ) ( _ ) ( _ ) ( _ ) 8 ( _ ( _ ) ) ( _ ) P i (09) t P i P i (09) t P i function and find their intersection P i P i 1_ (09) The population will reach half of t its original value in approimately t 9 96 years Module 1 716 Lesson 1 _ 1 Image Credits: prudkov/ Shutterstock Focus on Patterns MP8 When solving an equation involving a variable eponent, suggest that students try to structure the solution so that they are solving an equation of the form b c y If c b, then y; if c b, then they should solve by graphing Eercise Depth of Knowledge (DOK) Mathematical Practices Strategic Thinking MP Modeling Strategic Thinking MP Logic Using Graphs and Properties to Solve Equations with Eponents 716
AVOID COMMON ERRORS Students may be confused by complicated equations that involve variable eponents as well as additional factors Remind them to first apply the properties of equality to isolate the number with the variable eponent, then use the Equality of Bases Property to solve 18 An animal reserve has 0,000 elk The population is increasing at a rate of 11% per year How long will it take for the population to reach 80,000? The model for population is P (t) 0,000 (111) t, where t is the time in years and P (t) is the population in year t To find when the population is 80,000, solve P (t) 80,000 for t 80,000 0,000 (111) t Using a calculator graph each side of the equation as a function and find their intersection t 66 The population will reach 80,000 in approimately 66 years 19 A lake has a small population of a rare endangered fish The lake currently has a population of 10 fish The number of fish is increasing at a rate of % per year When will the population double? How long will it take the population to be 80 fish? The model for population is P (t) 10 (10) t, where t is the time in years and P (t) is the population in year t Solve P (t) 0 for t 0 10 (10) t Using a calculator graph each side of the equation as a function and find their intersection t 1767 The population of the fish will double in 18 years To find when the population will be 80, you can solve P (t) 80 for t Alternatively, note that 80 10 8 10 This corresponds to the PAGE 6 population doubling three times, from 10 to 0, from 0 to 0, and from 0 to 80 The population will be 80 in years ( 18) 0 Tim has a savings account with the bank The bank pays him 1% per year He has $000 and wonders when it will reach $00 When will his savings reach $00? The model is S (t) 000 (101) t Solve S (t) 00 for t 00 000 (101) t Using a calculator graph each side of the equation as a function and find their intersection Graphing f () and g (), we get the point of intersection (9168, 10) Rounding up and considering interest is calculated yearly, it will take Tim years Module 1 717 Lesson 1 717 Lesson 1 1
1 Tim is considering a different savings account that pays 1%, but this time it is compounded monthly (When interest is compounded monthly, the bank pays interest every month instead of every year The function representing compounded interest is S (t) P (1 + r ) nt n, where P is the principal, or initial deposit in the account, r is the interest rate, n is the number of times the interest is compounded per year, t is the year, and S (t) is the savings after t years) How many years will it take Tim to earn $00 at this bank? Should he switch? The model is S (t) 000 (1 + _ 001 1 ) 1t or S (t) 000 (10008) 1t Solve S (t) 00 for t 00 000 (10008) 1t Using a calculator, graph each side of the equation as a function and find their intersection t 9 Graphing f () and g (), we get the point of intersection (970, 10) is approimately years Both accounts will reach $00 in about years Switching won t make much difference Lisa has a credit card that charges % interest on a monthly balance She buys a $00 bike and plans to pay for it by making monthly payments of $100 How many months will it take her to pay it off? Assume the first payment she makes is charged no interest because she paid it before the first bill Her first payment is $100 At that time she owes $100 plus interest or $10 The second month she pays $100 and the third month she pays the rest It takes her three months to pay it off You do not have to solve an eponential because % is not a very high interest Analyze Relationships A city has 17,000 residents The population is increasing at the rate of 10% per year a You want to make a prediction for how long it will take for the population to reach 00,000 Round your answer to the nearest tenth of a year b Suppose there are 0,000 residents of another city The population of this city is decreasing at a rate of % per year Which city s population will reach 00,000 sooner? Eplain On parts a and b using a calculator graph each side of the equation as a function and find their intersection a 00,000 17,000 (11) 7 The population will reach 00,000 in approimately 7 years b 00,000 0,000 (97) 1 1 < 7 The second city s population will reach 00,000 sooner AVOID COMMON ERRORS Some students may be unsure how to raise a fraction to a power Remind them that both the numerator and the denominator must be raised to the same power CURRICULUM INTEGRATION Encourage students to research applications of eponential functions They should consider applications in science and business as well as uses in other math courses Focus on Reasoning MP As students solve real-world problems involving time, have them make predictions before calculating their results Write the predictions on the board, then compare them to the solutions found algebraically or graphically Encourage students to improve their predictions by analyzing whether their predictions tend to be too high or too low and by considering how they can change their estimation methods Module 1 718 Lesson 1 Using Graphs and Properties to Solve Equations with Eponents 718
VISUAL CUES Have students create posters as visual reminders of how to solve equations involving eponents Remind students to include eamples as well as step-by-step procedures JOURNAL In their journals, have students eplain how to use the Equality of Bases Property to solve an equation with a variable eponent HOT Focus on Higher Order Thinking Eplain the Error Jean and Marco each solved the equation 9 () 79 Whose solution is incorrect? Eplain your reasoning How could the person who is incorrect fi the work? Jean Marco 9 () 79 9 () 79 1_ ( 9) 9 () ( 1_ 9) 79 () 79 81 + 79 6 6 Jean is completely correct and Marco could correct his work as follows: Marco 9 () 79 () 79 + 79 6 + 6 He substituted for 6 instead of + 6, which yields Critical Thinking Without solving, identify the equation with the greater solution Eplain your reasoning 1_ ( ) () 1_ ( ) (9) The equation 1_ () has a greater solution Since the values of the powers of increase less quickly than the values of the powers of 9, the value of in 1_ () will be greater than the value of in 1_ (9) Module 1 719 Lesson 1 719 Lesson 1 1
Lesson Performance Task A town has a population of 78,918 residents The town council is offering a prize for the best prediction of how long it will take the population to reach 100,000 The population rate is increasing 6% per year Find the best prediction in order to win the prize Write an eponential equation in the form y ab and eplain what a and b represent Write an eponential equation Let y represent the population and represent time in years a represents the initial population 78,918 b represents the rate of increase in the population per year y 78,918 (1 + 006) Substitute the target population for y: 100,000 78,918 (1 + 006) Write the Substitute the target population for y: 100,000 78,918 (1 + 006) as functions f () 100,000 g () 78,918 (1 + 006) Use the intersect feature on a graphing calculator to find the point of intersection The point of intersection is (06, 100,000) The population will reach 100,000 in just over years Image Credits Jim West/ Alamy Images PAGE 66 Focus on Modeling MP Before students write an equation for the situation in the Lesson Performance Task, discuss how they know that the base to be raised to a power in the eponential equation is 106 and not 006 Have them consider What factor multiplied by the population makes the number 6% greater? Then ask what the base would be if the population were decreasing by 6% per year Students should recognize that it would be 1 006 09 Discuss what a graph showing each growth rate would look like Focus on Technology MP As students use their graphing calculators to graph the two functions and find their intersection, remind them to adjust the viewing window so that the intersection is shown clearly Focus on Communication MP Have students share their reasons for why the point where the graphs of the right- and left-hand sides of the equation intersect is the solution Module 1 70 Lesson 1 EXTENSION ACTIVITY Have students research the current population of their community or state and the rate at which it is growing or decreasing Then have students write an eponential equation in which y represents the population and represents time in years Finally, have students choose a future population size and predict when the population will reach that size Scoring Rubric points: Student correctly solves the problem and eplains his/her reasoning 1 point: Student shows good understanding of the problem but does not fully solve or eplain his/her reasoning 0 points: Student does not demonstrate understanding of the problem Using Graphs and Properties to Solve Equations with Eponents 70