Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals
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1 Secondary Two Mathematics: An Integrated Approach Module 3 - Part One Imaginary Number, Exponents, and Radicals By The Mathematics Vision Project: Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius In partnership with the Utah State Office of Education "2013"Mathematics"Vision"Project" "MVP" In"partnership"with"the"Utah"State"Office"of"Education""" Licensed(under(the(Creative(Commons(Attribution4NonCommercial4ShareAlike(3.0(Unported(license." 1
2 Task 3.1 Maximum Power In this task, we will be experimenting with some useful properties of exponents. To start, let s make sure we remember exactly how exponents work. In much the same way that multiplication is simply repeated addition, exponents are a tool for expressing repeated multiplication. So for example = = = = a. Rewrite the following expression in its expanded form: b. Now use your expanded form to rewrite the expression as a single base raised to a single power: 2. Repeat the process above to simplify the following expressions: Can you identify a shortcut that would have allowed you to simplify these expressions more quickly? 4. Use your answer to question 3 to complete the following mathematical property: The Product of Powers Rule: bb xx bb yy = bb 5. Would this property help you simplify the following expression? Why or why not? Use this new property to simplify each of the following expressions: xx 3 xx 6 mm 2 mm 3 mm 2
3 There are more exponent properties we can discover by thinking about expanded form. For example, we can consider what happens when we raise an exponential expression to a further power, like in this example: (44 22 ) 33 = = Repeat the process above to simplify the following expressions. (2 3 ) 4 (7 4 ) 5 9. Can you identify a shortcut that would have allowed you to simplify these expressions more quickly? 10. Use your answer to question 9 to complete the following mathematical property: The Power of Powers Rule: (bb xx ) yy = bb 11. Use this new property to simplify each of the following expressions: (5 5 ) 12 (xx 7 ) 3 (xx 6 yy 2 ) 4 Here s another use for expanded form: = = Repeat the process above to simplify the following expressions:
4 13. Can you identify a shortcut that would have allowed you to simplify these expressions more quickly? 14. Use your answer to question 13 to complete the following mathematical property: The Quotient of Powers Rule: bb xx bbyy = bb 15. Use this new property to simplify each of the following expressions: xx 5 xx 10 yy 5 xx 2 xx Use all of the rules you have learned in this module to simplify the following expression: xx9 bb 2 3 xx 4 bb 5 4
5 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals READY, SET, GO Name Period Date READY Topic: Evaluating Functions Graphically 1. Sketch a graph of each of the following functions: a. f(x) = 2 x b. g(x) = 3 x 2. Use the above graphs to evaluate each of the following. If you are unsure of an exact answer, estimate. a. f(0) = e. g(1) = b. f(1) = f. g(2) = c. f(2) = g. g(1.5) = d. f(1.5) = h. g(0.5) = Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 5
6 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals SET Topic: Properties of Exponents Rewrite each of the following expressions in its simplest exponential form using the Product of Powers, Power of Powers, and Quotient of Powers Rules. GO Topic: Imaginary Numbers Simplify each of the following expressions. 6
7 Task 3.2 Be Rational We are very familiar with the idea of exponents, which tell us how many copies of a particular number to multiply together (or if the exponent is negative, how many times to divide by that number). Today, we will be exploring what happens when our exponent isn t a whole number, but instead a rational number (i.e. a fraction). 1. Use the provided graph in order to calculate the following values. If you cannot determine the exact value from the graph, estimate. ff(xx) = 2 xx ff(0) = ff(1) = ff(2) = ff 1 2 = ff 3 2 = 2. Using a calculator and the above function, calculate each of the following values (round to the 2 nd decimal place): ff 1 2 = ff 3 2 = ff 9 4 = Your calculator has no trouble providing you with answers for the above problems, but you would probably struggle to explain why the answers make sense. If exponents tell us how many copies of a number to multiply together, how exactly do we multiply together 0.5, 1.5, or 2.25 copies of a number together? Clearly, exponents are more complicated than we thought, so let s try to think about this problem a different way. 3. Assuming that each of the following patterns are geometric (they change by multiplying), find the common ratio and fill in the missing values
8 4. Now we ll try working with some related exponential equations. How might you fill out the missing columns in the following table? xx What s Happening? What s Really Happening? ff(xx) = 1 9 xx From the last two problems, we can see that the following two things are true: = 3 and 9 = 3 Based on these two statements, what conclusion might you make? 6. Try filling out the following table the same way: xx What s Happening? What s Really Happening? ff(xx) = 1 8 xx 7. Based on the above table, how might you finish the following statement? = In this task, we have been working with rational exponents, where in the past all of our exponents have been integers. When faced with a rational exponent, we can handle it the following way: bb xx yy = = x tell us y tells us 8
9 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals READY, SET, GO Name Period Date READY Topic: Finding a Prime Factorization In past math classes, you may have built a factor tree to help you find the prime factorization of a number. For instance, if you wanted the prime factorization of the number 54, you might build a tree like this one: Each number in a box is a composite number, and each circled number is a prime number (meaning it can only be divided by itself or by 1). Once every "branch" of your tree ends with a circle, you know you are finished. Using this tree, we can see that the prime factorization of 54 is......because there was one 2 and three 3's. 54 = 2 3³ Build factor trees and write the prime factorization of each of the following numbers: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 9
10 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals SET Fill in the tables of values and find the factor used to move between whole number values, F w, as well as the factor, F c, used to move between each column of the table. 7. x y 4 16 d. F w = e. F c = F w F w 8. x y 5 15 d. F w = e. F c = F w F w Rewrite each of the following as the product of whole numbers: Ex: GO Topic: Square Numbers Make a list of the first 15 square numbers. The list has been started for you: 12. 1, 4, 9, 10
11 Task 3.3 I m Fluent in Exponents Today s goal is to practice with rational exponents and strengthen your mathematical fluency, the same way you might grow more fluent in a spoken language. 1. Without using a calculator, determine the value of each of the following (your answers should be whole numbers): = = = = 2. It is often useful to be able to think of mathematical expressions in different forms. Rewrite each of the following exponential expressions in their matching radical forms: EEEE: oooo ( 7) = xx 7 2 = (7yy) 4 9 = 5 3bb 2dd = 3. Now try going the other way: 11 aa 3 h = = 2xx = 4. How might you change these to exponential form? 4 aa 3 bb 8 2 = xx 1 yy 7 zz 3 = 5. All of the exponent rules we learned apply to rational exponents the same way they applied to integer exponents. Use exponent properties to simplify each of the following: xx 1 7 xx 3 7 = xx 1 2 xx 2 3 = (xx 1 7) 3 2 = (xx 2 5) 5 2 = 11
12 SECONDARY MATH II // MODULE Imaginary Numbers, Exponents and Radicals 3.3 READY, SET, GO Name Period Date READY Topic: Square Factors Rewrite each number as the product of two factors, one of which is a square number. Try to use the largest square factor possible: Ex: 48 = x³ SET Topic: Radical notation and radical exponents Each of the expressions below can be written using either radical notation, or rational exponents. Rewrite each of the given expressions in the form that is missing. Radical Form 9. Exponential Form " "
13 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals GO Topic: Exponent Rules Simplify the following expressions using the product, power, and quotient rules: 16) 2 7 x x 17) x ) 5 2x 19) ) 0 8x 21) ) 3 2x 8x 4 23) (-3) 4 24) 7 xy xy ) x 3x x 26) 3st ) 2 7 3m n m 5 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 13
14 Task 3.4 Simple Is as Simple Does We have often seen that multiple equations can represent the same function. This is because these equations are really just different forms of the same equation. For example yy = and yy = 33(xx 22) are just slope-intercept and point-slope form equations of the same line. The same idea can work for numbers, like how 1111, , and are all just alternative ways of writing the number 4. Today we will look for a similar relationship in radicals. 1. Use a calculator to approximate the value of each of the following radicals. Round your answers to the first decimal place: = = 8800 = A quick check with a calculator shows us that even though the numbers looked different, they are in fact the same. Today s goal is to figure out exactly why these values are the same, and then to extend that idea to include more complicated radicals ( radical is a fancy word for square roots, cube roots, etc.). 2. See if you can simplify each of the following expressions in any way: = = aa xx bb xx = 4 1/2 5 1/2 = Rational exponents are equivalent to radicals, so a rule that works in exponential form, like 44 11/ /22 = /22, must also work in radical form, like = This creates a brand new rule for dealing with radicals: aa bb = aa bb Using this same property, we can demonstrate that the above answers must be equal. 3. Rewrite 28 as the product of two factors, one of which is a square number. Then use the above property to simplify 28: 4. Try simplifying each of the following square roots the same way we simplified = 75 = 72 = 24 = 14
15 Another way you might simplify radicals is by using prime factorization. For example, if I wanted to calculate 2222, I would first build a factor tree and find the prime factorization. Then I would rewrite and solve my problem using that prime factorization: 2222 = = = Try simplifying the same square roots using this new method. You might find the last one a little trickier: 18 = 75 = 72 = 24 = 6. By using the prime factorization, we are able to simplify more complex radicals. Try simplifying each of the following. Remember, the inverse of a 3 rd root is a third power, the inverse of a 4 th root is a 4 th power, etc = 80 3 = 162 = 15
16 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals READY, SET, GO Name Period Date READY Topic: Combining Like Terms Simplify each of the radicals SET Topic: Simplifying radicals, imaginary numbers When simplifying expressions, we can only combine like terms. Simplify each of the expressions below. You may need to simplify some radicals in order to identify like terms. 10. x + 2x 11. 2x + 3x + 5y Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 16
17 SECONDARY MATH II // MODULE 3 Imaginary Numbers, Exponents and Radicals GO Topic: Solve Quadratic Equations Identify a pattern and then find the next three terms of the sequence: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 17
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