extending the definition of exponents © math as a second language all rights reserved next #10...

Extending the

Definition of

Exponents© Math As A Second Language All Rights Reserved

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#10

Taking the Fearout of Math

2-8

In illustrating the use of positive integer exponents, we have chosen

several different applications including how to compute the number of outcomes when a coin is flipped a certain number

of times.

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© Math As A Second Language All Rights Reserved

This was a good application to use because a coin can only be flipped a whole number of times. It makes no

sense to talk about what happens, for example, when a coin is flipped

“negative 2” times.

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However, there are other applications in which we have occasion to use

exponents that include other integers (that is, 0 and the negative integers).

One such application is when we want to extend the use of exponential notation to represent the powers of 10 when we

deal with decimals numbers.

When all we had were the positive integers, the denominations were

represented as shown below…

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Place ValueNotation

ExponentialNotation

10,000 104

1,000 103

100 102

10 101

As we have mentioned in previous lessons, definitions and rules are often

based on things we would like to be true or patterns we would like to continue.

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Notice that in the table, each number is divided by 10 to get to the number

below it.”

Place ValueNotation

ExponentialNotation

10,000 104

1,000 103

100 102

10 101

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next Therefore, if we want this pattern to continue, the next entries would have to be

based on the facts that 10 ÷ 10 = 1,

1 ÷ 10 = 1/10; 1/10 ÷ 10 = 1/100 = 1/102 etc.

Thus, the extension of

the first column

would have to look like.

Place ValueNotation

ExponentialNotation

10,000 104

1,000 103

100 102

10 101

11/10

1/100

1/1000

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next Again, assuming that we want the same pattern to continue, we notice that as we read down the second column, the exponent decreases by 1 from its value

in the row above.

In other words, the sequence of exponents would look like…

104, 103, 102, 101, 100, 10-1, 10-2, 10-3, etc…

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Our chart would then look like…

Place ValueNotation

ExponentialNotation

10,000 104

1,000 103

100 102

10 101

100

10-1

10-2

11/10

1/100

1/1000 10-3

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So if we want the chart to continue in the same way, what must be true is that…

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100 = 1

10-1 = 1/10 1

10-2 = 1/100 = 1/102

10-3 = 1/1000 = 1/103

In other words, 10n and 10-n are reciprocals of one another. That is, if n is any

positive integer 10-n = 1 ÷ 10n

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Notice that defining 100 to be 1 is

reasonable in the sense that if n is any

positive integer, 10n is a 1 followed by n zeroes. So it seems natural that if the

exponent was 0, the number would be a 1 followed by no 0’s (that is, 1).

Notes

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We know that to multiply a decimal number by 1,000, we move the decimal point

3 places to the right, and to divide a decimal number by 1,000 we move the

decimal point 3 places to the left. In that context, it seems natural that if 10+3 tells us to move the decimal point 3 places to the right, that 10-3 should tell us to move the

decimal point 3 places to the left.

Notes

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However, mathematicians have a better reason for defining integer exponents

the way we do. Without going into the reasons behind the decision, it turns out that there are good reasons to make sure that the rules that govern the arithmetic of non-zero whole number exponents should also apply to any other exponents as well.

Notes

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With this in mind, let’s see how we would have to define b0. We already know

that for positive integer exponents

bm × bn = bm+n.

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So if we were to let n = 0, the rule would become… bm × b0 = bm+0

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Since m + 0 = m, this would mean that…

bm × b0 = bm

If we now divide both sides of the above equation by bm, we see that b0 = 1.1

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1 If b = 0 then bm is also equal to 0. However, we are not allowed to divide by 0. Therefore, we have to add the restriction that b ≠ 0.

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Another way of saying this is to observe that since bm × b0 = bm, this

equation tells us that b0 is that number which when multiplied by bm yields bm as the product, and this is precisely what it

means to multiply a number by 1.

Note

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Note

By way of review, the reason we had to add the restriction that b ≠ 0 when defining b0 is

based on the fact that any number multiplied by 0 is 0. Notice that if we replace b by 0 in the

equation bm × b0 = bm, we obtain the result that 03 × 00 = 03.

Since we know that 03 = 0, this says that 0 × 00 = 0. But, any number times 0 is 0!

Therefore 00 can equal any number. When this happens, we say that the value of 00 is indeterminate; meaning that it can be any

number.

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For example, suppose that for some “strange” reason we wanted to define

00 to be 7. If we replace 00 by 7, 0 × 00 = 0becomes 0 × 7 = 0, which is

a true statement.

For b ≠ 0, we have defined b0 to be 1. It does not have to be defined to equal 1, but

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 the rule for multiplying like bases.

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Students often feel that b0 should equal 0 because there are no factors of b. What

we can tell students when this happens is that there is nothing wrong with thinking that it should be 0, but if we let it equal 0,

we lose the use of the rules that make computing so convenient.

Key Point

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Perhaps it will make it easier for students to feel comfortable with our defining b0 to be equal 1 if we point out that since bn × 1 = bn to the exponent, 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. As we have already discussed, this is especially easy to visualize when b = 10. Namely, in this case 10n is a 1 followed by n zeroes.

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

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In the case where n = 0, 20 would be the number of outcomes that can occur if we don’t flip the coin at all.

There is ONE outcome - the current state of the coin.

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One reason that mathematicians prefer to work abstractly is that physical

models do not always exist, and even when they exist, they might not make

sense in some instances.

For example, we cannot usethe “flipping a coin” model to explain the

use of negative exponents. However, as we have explained earlier, negative exponents

make sense when we are dealing with powers of 10.

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The “powers of 10” model gives us a clue as to why b-n = 1/b

n might still be a correct rule to use even if b ≠ 10. More

specifically, we have already given a plausible explanation as to why this is true in the case where b = 10. So based on the

fact that 10-n = 1/10n, we might want to

conjecture that for any base b, b-n = 1/bn.

A more consistent reason might be that we still want bm × bn = bm+n to be true even when m and/or n is a negative integer.

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To this end we already know that when we multiply like bases we add the

exponents, and we also know that for any number, n, n + -n = 0.

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Therefore… bn × b-n = bn + -n = b0

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And since we have already accepted the definition that b0 = 1, it follows that…

So if now we divide both sides of the equality bn × b-n = 1 by bn, it follows that

b-n = 1 ÷ bn.

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bn × b-n = bn + -n = b0 = 1

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More generally, if m and n are any integers and b and c are any numbers,

then it is still true that…

(6) (b × c)n = bn × cn

(2) b0 = 1 provide that b ≠ 0; 00 is indeterminate2

(3) bn = 1 ÷ b-n

(4) bm ÷ bn = bm-n

(5) (bm)n = bm×n

(1) bm × bn = bm+n

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2 Whenever an exponent is 0 or negative the base b cannot equal 0.

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nextnext To help you internalize the rules let’s suppose that you wanted to rewrite the

expression 102 × 10-5 as a decimal number.

By Rule (1) (with b = 10, m = 2 and n = -5) 102 × 10-5 = 10-3 = 0.001. As a check, notice

that…

102 × 10-5 = 102 × 1/10-5

= 100 × 1/100,000

= 100/100,000

= 1/1000 = 0.001

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In the same way that something may increase exponentially, another thing might decrease exponentially. One such example is in terms of radioactive decay where we

talk about the half-life of a radioactive substance. The half life is the amount of

time it takes for the substance to “shrink” to half its present weight (mass).

An Enrichment Noteon

Integer Exponents

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next Rather than talk about radioactivity, let’s suppose instead that you have received a gift of $128 and you don’t

want to spend it all at once. Instead you decide to spend half of it now and then each of the following weeks, you will

spend half of what is left.

So the first week you spend half of the $128, leaving you with $64; the next week

you spend half of $64, leaving you with $32; the next week you would spend half

of the $32, leaving you with $16, etc.

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next Generally, if P denotes the present amount, a week later the amount left is

1/2 × P, after the second week the amount left is 1/2 × (1/2 × P) or (1/2)2× P, and after

the third week the amount left is

(1/2)3 × P., etc.

However, in order not to have to use fractional notation we may rewrite an

expression such as (1/2)3 × P in the form 2-

3 × P. After n weeks the amount left could then be expressed as 2-n × P.

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Until now there seems to be no pressing reason to define fractional exponents.

However, let’s suppose that we knew that the cost of living was increasing at a rate of 4% per year. That means what costs $1.00 this year will cost $1.04 next year. So if C represents the cost of living this year, the

cost of living next year will be 1.04 × C

An Enrichment Noteon

Fractional Exponents

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Thus, the next year it will be 1.04 × C, not C, that is increasing by 4% a year. So at the end of the second year the cost of living

is 1.04 × (1.04 × C), or 1.042 × C. And in a similar way at the end of the third year the

cost of living is 1.043 × C. More generally, at this rate at the end of n years the cost of

living would be 1.04n × C.

However, if we wanted to know the cost of living 6 months (i.e., 1/2 year) from now,

using the above formula, it would be given by 1.041/2 × C .

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This motivates us to try to define 1.041/2

More specifically, if we still want it to be true that bm × bn = bm+n, we may replace m and n by 1/2 to obtain…

b1/2 × b1/2 = b1/2 + 1/2 = b1 = b.

In other words, b1/2 is that number which when multiplied by itself is equal to b.

By definition of the square root (that is, the square root of a given number is the

positive number which when multiplied by itself is equal to the given number), it

means that b1/2 = √b .

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As a check that this definition is plausible, we can use our calculator and

representing 1/2 as 0.5, we see, for example, that 9.0.5 = 3, and as a check, we know that

3 x 3 = 9. Hence, 9.0.5 is the square root of 9.

In a similar way, we see that 1.041/2 = 1.01980…, and this, in turn tells us that if something costs $100 today,

6 months from now it will cost 1.01980… × $100 or, to the nearest cent,

$101.98.

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This discussion completes our arithmetic

course.


Algebra is next.

We hope you will join us for our algebra course.

extending the definition of exponents © math as a second language all rights reserved next #10...

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