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

The Game of Algebraor

The Other Side of Arithmetic

The Game of Algebraor

The Other Side of Arithmetic

© 2007 Herbert I. Gross

byHerbert I. Gross & Richard A. Medeiros

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Lesson 7Part 2

The Arithmetic of

Exponents

The Arithmetic of

ExponentsWhen the exponents are not whole

numbers!© 2007 Herbert I. Gross

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There are times when exponents must be whole numbers. For example, we cannot

flip a coin a fractional or a negative number of times. However suppose you

have a long term investment in whichthe interest rate is 7% compounded

annually. Knowing the present value of the investment, it makes sense to ask what the value of the investment was,

say, 3 years ago.

© 2007 Herbert I. Grossnext

Moreover we might even want to invent exponents that are not integers. For example suppose the cost of living

increases by 6% annually. We might want to know how much it increases by every

6 months (that is, in 1/2 of a year). It might come as a bit of a surprise, but as we shall see later in this presentation the

answer is not 3%)


A device that is often used in mathematics is that when we extend a definition we do it in a way that preserves the original definition. In the case of exponents we like the properties that were discussed in Part 1 of this presentation, namely…


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bm × bn = bm+n

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bm ÷ bn = bm-n

(At this point we have not yet defined negative exponents so we have to remember that so far this property assumes that m is greater than n;

that is, m – n cannot be negative.)

(bm)n = bmn (bn × cn) = (b ×c)nnext

With this in mind, let's look at an expression such as 20. Notice that so

far we have only defined 2n in the case for which n is a positive integer. 0 is considered to be neither positive nor

negative. Thus, we are free to define 20 in any way that we wish.


Thinking in terms of flipping a coin, it seems that 20 should represent the

number of possible outcomes if a coin is never flipped. So we might be tempted to

say that 20 = 0 because there are no outcomes. On the other hand, the fact

that nothing happens is itself an outcome, so perhaps we should define

20 to be 1.


However, how we choose to define 20 will be based on

the decision that we would like…

bm × bn = bm+n to still be correct even

when one or both of the exponents are 0.


So suppose for example that we insist that 23 × 20 = 23+0. Since 3 + 0 = 3, this would mean that…


Another way to obtain this result is to divide both sides of the equation 23 × 20 = 23 by 23 to obtain…

This tells us that 20 is that number which when multiplied by 23 yields 23 as the product, and this is precisely what it means to multiply a number by 1. That is 20 must equal 1.

we see that 20 = 1. nextnextnext

23 × 20 = 23

23 231 1

This same result can be obtained algebraically without the use of exponents

by replacing 23 by 8 and 20 by x …


And dividing both sides by 8, we obtain…

Since x = 20 we see that 20 = 1.nextnextnext

x = 1

23 × 20 = 238x8

=

More generally, if we replace 2 by b


Key Point

23 × 20 = 23 + 0

and cancel bn from both sides of the equation,

b

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nb

and 3 by n,

nb

we see that for any non-zero number b, b0 = 1

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bn bn1

• The reason we must specify that b ≠ 0 is based on the fact that any number

multiplied by 0 is 0. More specifically, if we replace b by 0 in the equation

b3 × b0 = b3, we obtain the result that 03 × 00 = 0. Since 03 = 0, this says that

0 × 00 = 0; and since any number times 0 is 0, we see that 00 is indeterminate, where by indeterminate, we mean that it can be any

number. © 2007 Herbert I. Gross

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Note

• This is similar to why we call 0 ÷ 0

indeterminate. Namely 0 ÷ 0 means the set of numbers which when multiplied by 0 yield 0 as the product; and any number times 0 is

equal to 0.


Note

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• We say such things as 6 ÷ 3 = 2 when, in

reality, we should say that 6 ÷ 2 denotes the set of all numbers which when multiplied by 2 yield 6 as the product. However since there is only one such number (namely, 3) there is no

harm in leaving out the phrase the “set of numbers”.

• The time that it is important to refer to the “set of numbers” is when we talk about

dividing by 0. For example, b ÷ 0 denotes the set of all

numbers which when multiplied by 0 yield b as the product. Since any number

multiplied by 0 is 0, if b is not 0, then there are no such numbers, and if b is 0 the set

includes every number.© 2007 Herbert I. Gross

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Note

• For example, suppose that for some “strange” reason we wanted to define 00 to be 7. If we replace 00 by 7, it becomes …


Note

0 × 00 = 07

which is a true statement.

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We are not saying if b ≠ 0 that b0 has to be 1. Rather what we are saying is that if we

don't define b0 to equal 1, then we cannot use the rule, bm × bn = bm+n if either m or n is equal to 0. In other words by electing to let b0 = 1, we are still allowed to use this rule.


Key Point

For example if we were to let 20 = 9, the equation 23 x 20 = 20 would lead to the false statement 23 x 20 = 23; that is it

would imply that 8 x 9 = 8.

Note

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• There are other motivations for defining b0 to be 1.


Note

For Example

If we write bn as 1 x bn, then n tells us the number of times we multiply 1 by b. If we don't multiply 1 by b, then we still have 1.

This is especially easy to visualize when b = 10. Namely, in this case 10n is a 1 followed

by n zeros. Thus 100 would mean a 1 followed by no 0’s which is simply 1.

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In talking about an interest rate of 7% compounded annually. Namely, if

(1.07)n denotes the value of $1 at the end of n years, n = 0 represents the value of the dollar when it is first invested; which at

that time is still $1. In this context ($1.07)0 = $1.


Note

Using the same approach as above a clue to one way in which we can define bn in the case that n is a negative integer can be seen in the answer to the following

question.


How should we define 2-3 if we want Rule #1 to still be correct even in the case of

negative exponents?

Practice Question 1

Hint: 3 + -3 = 0next

© 2007 Herbert I. Gross next

Solution Answer: 2-3 = 1 ÷ 23 (= 1/23)

In order for it to still be true that bm × bn = bm + n, it must be that 23 × 2-3 = 23 + -3 = 20 = 1.

The fact that 23 × 2-3 = 1 means that 23 and 2-3 are reciprocals; that is, 2-3 = 1 ÷ 23.

Another way to see this is to start with 23 × 2-3 = 1 and then divide both sides by 23 to obtain that 2-3 = 1/23. If we now replace 2 by b and 3 by n, we obtain the more general

result that if b ≠ 0 and if n is any integer (positive or negative), b-n = 1 ÷ bn

Even though the exponent is negative, 2-3 is positive. It is the reciprocal of 23. What

is true is that as the magnitude of the negative exponent increases, the number

gets closer and closer to zero. For example, since 210 = 1,024; 2-10 = 1/1024 or in decimal form approximately 0.000977,

and since 220 equals 1,048,576, 2-20 is approximately 0.0000000954


Important Note

In fact the property bm ÷ bn = bm-n gives us yet another way to visualize why we define b0 to equal 1. Namely if we choose m and

n to be equal, we may replace m by n in the above property to obtain…

bn ÷ bn = bn-n = b0

And since bn ÷ bn = 1 (unless b = 0 in which case we get the indeterminate form 0 ÷ 0)

we may rewrite the above equality as 1 = b0.


Note

If we define b0 to equal 1 (unless b = 0, in which case 00 is undefined), and if we

also define b-n = 1 ÷ bn then the rules that apply in the case of positive whole

number exponents also apply in the case where the exponents are any integers.


Summary


For what value of n is it true that 42 ÷ 45 = 4n?

Practice Question 2

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Solution Answer: n = -3

If we still want bm ÷ bn = bm – n to apply, it means that 42 ÷ 45 = 42-5 = 4-3.

© 2007 Herbert I. Grossnextnext

Solution Answer: n = -3

If you prefer, instead, to “return to basics” use the definition for whole number

exponents to obtain…

42 ÷ 45 =42 45

= 4 × 4

4 × 4 × 4 × 4 × 4 1 1

43=

and since by definition 1/43 means the same thing as 4-3, the result follows.

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The display window of

the calculator displays 1 as the answer.

© 2007 Herbert I. Gross nextnextnext

To verify that 150 = 1, simply enter

“15”,

press the xy key;

enter“0”; press

the = key

=

0

9 8 7 +

6 5 4 -

3 2 1 ×

%

xy =

÷

On/off 1/x

0 .

1

1

5

5

xy

0

0

=

1

The calculator can also be used as a “laboratory” to verify some of the

properties about exponents.

For example

This is not really a proof of the formula b0 = 1.

Rather it is a demonstration that the formula is plausible. In other words, all we've

actually done is verified that 150 = 1. It tells us nothing about the value of b0 when b ≠ 15.


Of course, we can repeat the above procedure for as many values of b as we

wish, and each time we will find that b0 = 1. However, until we test the next value of b, all we have is an educated guess that the result will again be 1. Yet seeing that the

formula is correct in every case welook at tends to give us a better feeling

about the validity of the formula.


• No matter how small b is, as long as it isn’t 0, b0 will equal 1.

For example, (0.000000001)0 = 1. Sometimes even a calculator cannot

distinguish between 0.000000001 and 0; hence it might give 1 as the value of 00


Note

“0.001” now appears in the

display window.© 2007 Herbert I. Gross nextnextnext

Suppose you weren’t

certain that 10-3 = 1/103 = “0.001”. You could enter

“10”.

press the xy

key;

Enter “3”;

press the +/- key

=

0

9 8 7 +

6 5 4 -

3 2 1 ×

%

xy =

÷

On/off +/-

0 .

1

1

0

0

xy

3

-

=

If your calculator has a +/ – key (the “sign changing” key), you can even verify results

that involve negative exponents.

For example

press the = key

+/-

30.001

(Notice that 0.5 is the decimal

equivalentof the

reciprocal of 2. That is, 1 ÷ 2 = 0.5).

© 2007 Herbert I. Gross nextnextnext

If you enter “2” and press

the 1/x key, “0.5” appears

in thecalculator's

display window. =

0

9 8 7 +

6 5 4 -

3 2 1 ×

%

xy =

÷

On/off 1/x

0 .

2

2

=

Scientific calculators also have a reciprocal key. It looks like 1/x.

For example

1/x

0.5

Since 10-3 is the reciprocal of 103, another way to compute the value of 10-3 is to

compute 103 and then press the 1/x key.


Not all fractions are represented by terminating decimals. For example, if you use the calculator to compute the value of 3-2 , the answer will appear as “0.111111111”

which is a rounded off value for 1/9 or 3-2. In this sense, using the calculator will give

you an excellent approximation to the exact answer, but it might tend to hide the

structure of what is happening.

Caution

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As a practical application of negative exponents let's return to an earlier

discussion where we talked about the growth of, say, $10,000 if it was invested at

an interest rate of 7% compounded annually for 10 years. We saw that after ten

years, the amount of the investment, (A),would be given by…

A = $10,000(1.07)10


A companion question might have been to find the amount of money that would have had to have been invested 10 years ago at

an interest rate of 7% compounded annually in order for the investment to be

worth $10,000 today. In that case, the amount (A) would be given by…

A = $10,000(1.07)-10


That is, if we knew how much money there was at the end of a year, we would simply divide that amount by 1.07 in order to find what the amount was at the start of that year. Dividing by 1.07 ten times is the

same as dividing by (1.07)10 which is the same as multiplying by (1.07)-10.


The display will now show that rounded off to the nearest cent, $5,083.49 would

have had to have been invested 10 years ago in order for the investment to be

worth $10,000 today.© 2007 Herbert I. Gross

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1.07 xy 10 +/- =

To compute the value of 10,000(1.07)-10 using a calculator, try the following sequence of

key strokes…

× = 5,083.4910000

In fact on a year by year basis, the growth of $5,083.49 would have looked like…


10 years ago $5,083.499 years ago8 years ago7 years ago6 years ago5 years ago4 years ago3 years ago2 years ago1 year ago

Now

$5,439.33$5,820.09$6,227.49$6,663.42$7,129.86$7,628.95$8,162.97$8,734.38$9,345.79$9,999.99

Although fractional exponents are beyond our scope at this point, notice that they can be

motivated by asking such questions as “How much will the above investment be worth 6

months (that is, 1/2 year) from now?”. In that situation it makes sense to assume that if we

want our definitions and rules to still be obeyed, the formula should be…

A = $10,000(1.07)1/2 = A = $10,000(1.07)0.5


So to give a bit of the flavor of fractional exponents let's close this presentation with

what to do when the exponent is 1/2 .next

As an example, suppose we were given an expression such as 91/2. We know that

1/2 + 1/2 = 1. Therefore, if we want bm × b n

to still be equal to bm+n, then we would have to agree that …

91/2 × 91/2 = 91/2 + 1/2 = 91 = 9


In other words 91/2 is the number which when multiplied by itself is equal to 9. This is precisely what is meant by the (positive)

square root of 9 (that is, √9 ). In other words 91/2 = 3.

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The same result can be obtained algebraically by letting x = 91/2

© 2007 Herbert I. Grossnextnext

x = 9x

× x = 9x2

91/2 × 91/2 = 9

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• Based on our knowledge of signed numbers there are two square roots of 9,

namely 3 and -3. However since 91/2 is between 90 (=1) and 91 (=9), 91/2 has to be 3.


Note

• Notice that although 1/2 is midway between 0 and 1, 91/2 is not halfway

between 90 and 91.

90 = 1 91/2 = 3 91 = 9next

Applying our above discussion to (1.07)0.5, we may use the xy key on our

calculator to see that (1.07)0.5 = 1.034408…(i.e., (1.034408…)2 = 1.07) In other words

at the end of a half year the value has increased by a little less than half of 7%.

That is, each dollar is then worth $1.034408...


This concludes our discussion of the arithmetic of exponents. In the next

lesson, we will apply this knowledge to the topic known as scientific notation.

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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