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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 6

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Introduction to Exponents Introduction to Exponents b2b2 © 2007 Herbert I. Gross next n5n5

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Think for a moment about how you would answer the question…. How much is three 2's? © 2007 Herbert I. Gross next Although 6 is the answer that most people would give (and it's also the answer that is usually accepted as being correct), the wording of the question leaves much to be desired. next

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That is, most people tend to hear the question as if it were… How much is the sum of three 2's? © 2007 Herbert I. Gross next However there are times when the correct question is… How much is the product of three 2's? next

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© 2007 Herbert I. Gross next For example, it is indeed correct to take the sum of three 2's (2 + 2 + 2) to answer: How much would 3 pounds of candy cost, at $2 per pound? At other times it is the product of three 2's that gives the correct answer. For example, suppose we flip a penny, a nickel and a dime; and want to record the number of different ways in which the coins can turn up heads or tails. next

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© 2007 Herbert I. Gross next As we shall see below, the correct answer is given by 2 × 2 × 2 and not 2 + 2 +2. More specifically… Looking at only the penny, we realize that it can turn up either heads or tails. That is, there are just two possible outcomes. next or

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© 2007 Herbert I. Gross next In terms of a chart… Penny Outcome #1 Outcome #2 heads tails The labels Outcome 1 and Outcome 2 are used simply to count the possible outcomes. It doesn't mean that Outcome 1 is more likely or more desirable than Outcome 2. Note

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© 2007 Herbert I. Gross next If you toss a nickel once and a penny once, the nickel also can turn up in only two possible ways, namely, either heads or tails. So if you toss both coins, this will double the number of possible outcomes. That is, the nickel has to be either heads or tails; regardless of whether the penny turned up heads or tails. and or

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© 2007 Herbert I. Gross next Again in terms of a chart… Nickel Outcome #1 Outcome #2 heads tails Outcomes Outcome #3 Outcome #4 heads tails heads Penny tails

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© 2007 Herbert I. Gross next Notice however that in terms of outcomes, flipping 2 coins once is equivalent to flipping 1 coin twice. Seen in terms of the chart, flipping 1 penny twice yields… Flip 1 Outcome #1 Outcome #2 heads tails Outcomes Outcome #3 Outcome #4 heads tails heads Flip 2 tails

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© 2007 Herbert I. Gross next Dime heads tails tail NickelPenny heads tails heads tails heads tails heads tails heads tails heads tails heads tails Outcome #1 Outcome #2 Outcomes Outcome #3 Outcome #4 Outcome #1 Outcome #2 Outcome #3 Outcome #4 Outcome #5 Outcome #6 Outcome #7 Outcome #8 next After flipping the nickel once and the penny once, suppose we then flip a dime. The dime also can turn up only heads or tails. Adding this to the chart, we see that this again doubles the number of possible outcomes.

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© 2007 Herbert I. Gross next Dime heads tails tail NickelPenny heads tails heads tails heads tails heads tails heads tails heads tails heads tails Outcome #1 Outcome #2 Outcomes Outcome #3 Outcome #4 Outcome #5 Outcome #6 Outcome #7 Outcome #8 Dime tail NickelPenny

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© 2007 Herbert I. Gross next In the above illustrations, listing the possible outcomes and then counting them is a simple process; but one that can quickly become tedious. Namely, with each additional coin (or flip), we double the number of possible outcomes. In terms of actually listing these outcomes, it means that with each additional coin and its flip, we double the number of rows in our chart. The number of rows quickly becomes cumbersome, as we illustrate in the following example. How much is the product of sixteen 2's?

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© 2007 Herbert I. Gross next One way to answer the question is by the (tedious) process, whereby we first compute 2 × 2 to get 4; and then compute 4 × 2 to get 8, and ultimately (barring any computational errors), we compute the product of sixteen 2's to finally obtain 65,536 as our answer. How much is the product of sixteen 2's? 2 × 2 = 4× 4 = 8× 8 = 16× 16 = 32× 32 = 64× 64 = 128× 128 = 256× 256 = 512× 512 = 1,024× 1,024 = 2,048× 2,048 = 4,096× 4,096 = 8,192× 8,192 = 16,384× 16,384 = 32,768× 32,768 = 65,536 2 2323 2424 2525 2626 2727 2828 2929 2 10 2 11 2 12 2 13 2 14 2 15 2 16

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© 2007 Herbert I. Gross next For example, if we were to flip 16 coins (or one coin 16 times), and wanted to show all the possible outcomes using a chart; the chart would contain 65,536 rows. HHHHHHHHHHHHHHHH1HHHHHHHHHHHHHHHT2HHHHHHHHHHHHHHTH3HHHHHHHHHHHHHHTT4-------------------TTTTTTTTTTTTTTHH 65,533 TTTTTTTTTTTTTTHT 65,534 TTTTTTTTTTTTTTTH65,535TTTTTTTTTTTTTTTT 65,536

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© 2007 Herbert I. Gross next So just as we used an abbreviation (16 × 2) to represent the sum of sixteen 2's, we also use an abbreviation (2 16 ) to represent the product of sixteen 2's. With this as preparation, we can move on into algebra to generalize our computations. If b is any number and n is any positive whole number, we use the abbreviation b n to stand for the product of n factors of b. That is, b n means… b × b × b × b × b × b … × b n factors of b Definitions

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© 2007 Herbert I. Gross next We call b the base and n the exponent, and we read b n as b raised to the n th power or more briefly as b to the n th. (Exception… we usually read b 2 as b squared and b 3 as b cubed.) next So for example, 3 6 (which we read as 3 to the 6th or as 3 raised to the 6th power) is an abbreviation for 3 × 3 × 3 × 3 × 3 × 3 (or 729). Definitions In this example, 3 is the base and 6 is the exponent. next

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Don't confuse (for example) 6 3 with 3 6. 3 6 is the abbreviation for 3 × 3 × 3 × 3 × 3 × 3 (or 729), while 6 3 is the abbreviation for 6 × 6 × 6 (or 216). So while the order of the two factors in a multiplication problem does not affect the product (6 × 3 = 3 × 6); interchanging the exponent and the base clearly does. © 2007 Herbert I. Gross next Be Careful

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Another example… 2 16 = 65,536 while 16 2 = 256 © 2007 Herbert I. Gross next Be Careful Comment… Notice that a shortcut for finding, say, the sum of a hundred 2's is to add 2 one hundreds (that is, 100 × 2 = 2 × 100); there is no similar shortcut for finding the product of a hundred 2's (2 100 ). next

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© 2007 Herbert I. Gross next Technology To The Rescue The x y key on the calculator allows us to eliminate the tedium that is involved in the step-by-step computing of the product of sixteen 2's (2 × 2 × 2 × 2 × 2...) Thus, we could use a calculator to compute the value of 2 16.

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© 2007 Herbert I. Gross next If your calculator has a key that is labeled x y or y x you may compute the value of 2 16 quite quickly by the following sequence of keystrokes (but keep in mind that some calculators operate differently from others; so you may have to check the manual for your calculator if the following sequence of steps doesn't work). 2xyxy 16=

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The display window of the calculator will show the answer: 65,536 © 2007 Herbert I. Gross next 2xyxy 16= In words: enter 2; press the x y key; enter16; press the = key 987+ 654- 321× % xyxy = ÷ On/off 0. xyxy 1 = 2 6 2 1 6 65,536

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© 2007 Herbert I. Gross next Suppose you are using a calculator and the answer to a calculation happens to be 2,000,000,000,000,000. Most likely your calculator will not have the ability to display all 16 digits of the answer. Instead it would use exponential notation and display the answer in the form 2 × 10 15. A Note on Calculator Limitations

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© 2007 Herbert I. Gross next Thus, for example, if we try to use the x y key to compute the value of, say, 2 100 the calculator would display the answer in a form such as 1.2676506 × 10 30. Since each time we multiply 1.267506 by 10, we annex a zero; in place value notation 1.2676506 × 10 30 would be written as… A Note on Calculator Limitations 1,267,650,600,000,000,000,000,000,000,000.

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© 2007 Herbert I. Gross next However, the exact value of 2 100 is… A Note on Calculator Limitations but there is no way for the calculator to display all these digits. 1,267,650,600,000,000,000,000,000,000,000 Hence, it gives the answer in a rounded off form. 1,267,650,600,228,229,401,496,703, 205,376 next

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© 2007 Herbert I. Gross next The study of exponential growth goes well beyond flipping coins. For example, it plays a huge role in retirement planning… Suppose you have $10,000 invested in a retirement fund, and that every 7 years the amount of this investment will double. In 35 years, your investment will have grown to $320,000. A Financial Math Application

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© 2007 Herbert I. Gross next Perhaps the easiest way to follow this doubling process is shown in the following chart. Time Amount in your Retirement Fund Now After 7 years After 14 years After 21 years After 28 years $10,000 $20,000 $40,000 $80,000 $160,000 $320,000After 35 years

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© 2007 Herbert I. Gross next Comment… the doubling process doesnt depend on how much money you invest at the start. Whatever that investment was; it will double every 7 years. Since 35 is the 5th multiple of 7, in 35 years your investment will have doubled 5 times. More generally… If we let P (for principal) represent the amount of your original investment, after 35 years the amount of this investment P will have grown to: 32 × P (that is, 2 5 × P). next That is, after 35 years, your original investment of $10,000 is 2 5 (that is, 32) times what you started with…32 × $10,000 = $320,000.

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© 2007 Herbert I. Gross next Science and mathematics are, in themselves, neither good nor bad. The principles are the same whether they are applied to good things or to bad things. For example, the fact that 2 20 is greater than 1,000,000 tells us that if a person went out and helped 2 people; and then each of these 2 people went out and helped 2 other people, etc.; by the 20th link in this chain over one million people would have been helped; with nobody having to help more than 2 people! Discussion

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© 2007 Herbert I. Gross next Yet the same mathematics tells us that if a person infected 2 people with AIDS; and then each of these 2 people infected 2 other people with AIDS, etc.; by the 20th link in the chain over one million people would have been infected by AIDS, with no one having to infect more than 2 people.

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© 2007 Herbert I. Gross next Which way the mathematical result is used depends on society as a whole; not just on the scientist and mathematician. This is why the humanities, the social sciences and the physical sciences should be studied as a unified whole rather than in fragmented form.

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© 2007 Herbert I. Gross Most banks talk in terms of an annual rate of interest. For example, a bank may talk about an interest rate of 7% compounded annually. A 7% annual rate of interest means… that every $100 that is invested at the beginning of the year will have earned $7 by the end of that year. That is, every $1 invested now will have become $1.07 at the end of that year. In still other words, to find how much money you will have at the end of a year, multiply the starting amount by 1.07. next A Financial Math Application

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© 2007 Herbert I. Gross But the bank may well be offering not just a 7% interest rate, but a 7% interest rate compounded annually. next A Financial Math Application So here is a new term, compound interest. At the end of each year, interest earned that year is added to that years base investment, and this sum (original investment plus interest earned) will constitute the next years base.

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© 2007 Herbert I. Gross For example, suppose we invest $10,000 in an account, at the rate of 7% compounded annually. Let's see what the value of the investment will be 10 years from now. next First, to find the amount that is invested at the end of any year, we multiply the amount that was invested at the beginning of the year by 1.07. That is, every dollar at the beginning of the year is worth $1.07 at the end of the year.

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© 2007 Herbert I. Gross next $19,671.51 $10,000 × 1.967151 $10,000 × 1.07 10 Therefore… Since we want to find the value of the investment at the end of 10 years, we must multiply $10,000 by 1.07 ten times; that is, we must multiply $10,000 by 1.07 10. Using the X y key, the calculator shows us that 1.07 10 = 1.967151.

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© 2007 Herbert I. Gross In terms of using the calculator, we could show that 1.07 10 = 1.967151 by the following sequence of keystrokes… next 987+ 654- 321× % xyxy = ÷ On/off 0. xyxy 1 = 1 0 1 1 0.0 7.071.967151 Hence, the original $10,000 investment is now worth $1.967151 × 10,000 or $19,671.51 (or a little less than double its original value). ×1 1 0, 000 = 0000 19,671.51

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© 2007 Herbert I. Gross If our calculator doesn't have the x y key, we could instead have it compute the amount of our investment year by year. Rounding off the amounts to the nearest cent, we see that… At the end ofAmount InvestedYears Increase 0 years$10,000 1 year$10,000 × 1.07 = $10,700$700 2 years$10,700 × 1.07 = $11,449$749 3 years$11,449 × 1.07 = $12,250.43$801.43 4 years$12,250.43 × 1.07 = $13,107.96$857.53 5 Years$13,107.96 × 1.07 = $14,025.52$917.56 6 years$14,025.52 × 1.07 = $15,007.30$981.78 7 years$15,007.30 × 1.07 = $16,057.81$1050.51 8 years$16,057.81 × 1.07 = $17,181.86$1124.05 9 years$17,181.86 × 1.07 = $18,384.59$1202.73 10 years$18,384.59 × 1.07 = $19,671.51$1286.92 next

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Compound interest is computed on the amount that exists at the beginning of each year. The previous chart shows that we would have $19,671.51 in the bank at the end of the 10 years. That is, 7% interest compounded annually means, the investment almost doubles every 10 years. © 2007 Herbert I. Gross next Simple versus Compound Simple interest means that the interest is computed only on the original amount, during the entire ten years. So at a 7% simple interest rate, a $10,000 investment at the end of 1 year will be worth $10,700, and after 10 years it will have increased by 70%, to $17,000. next

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Original investment, $10,000. In comparison to 7% simple interest, which remained at $700 per year, the previous compound interest chart shows that the effective interest rate increases each year; so that by the tenth year it has risen to 12.87%... © 2007 Herbert I. Gross next Compound - beats - Simple 10 years$18,384.59 × 1.07 = $19,671.51$1286.92 At the end ofAmount InvestedYears Increase …and the rate will keep increasing year by year. next

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© 2007 Herbert I. Gross next This concludes our introduction to exponents, and in our next lesson we will discuss the arithmetic that is involved when we work with exponents. b n

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