Arithmetic of Positive Integer Exponents © Math As A Second Language All Rights Reserved next #10...

Arithmetic of Positive Integer

Exponents

© Math As A Second Language All Rights Reserved

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#10

Taking the Fearout of Math

28

× 24

Sometimes the symbolism we use in doing mathematics “goads” us into doing

things that are incorrect.

For example, seeing a plus sign in the expression…

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3/7 + 2/7

might tempt students to conclude that…

3/7 + 2/7 = 5/14

…even if they knew that 3 sevenths + 2 sevenths = 5 sevenths

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A similar problem might occur when beginning students are first asked to

compute such sums as 24 + 23.

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For example, seeing the plus sign, students might be tempted to add the two bases (2’s)

to obtain 4 and to add the two exponents

(4 and 3 to obtain 7; and

thus “conclude” that…

24 + 23 = 47

The proper thing to do, especially when you are in doubt, is to return to

the basic definitions.

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In this case, we know that 24 means

2 × 2 × 2 × 2 or 16, and that 23 means

2 × 2 × 2 or 8.

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24 23+2 × 2 × 22 × 2 × 2 × 2

816…and 24 is a great deal less than 47

(which is 4 × 4 × 4 × 4 × 4 × 4 × 4 or 16,384).

+

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= 24= 24

The key point to observe in

the expression 24 + 23 is that

the group of four factors of 2, and the group of three factors of 2 are

separated by a plus sign.

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Thus, we are not multiplying

seven factors of 2.

Key Point

However, had there been a times sign then we would have had seven factors of 2. In other words…

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24 23×2 × 2 × 22 × 2 × 2 × 2

816 ×

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= 27

×2 × 2 × 2 × 2 2 × 2 × 2

= 128

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Stated in words, when the product of four 2’s is multiplied by the product of

three more 2’s, the answer is the product of seven 2’s. Therefore…

(2 × 2 × 2 × 2) × (2 × 2 × 2) =

… leaving us with seven factors of 2 which equals 27.

(2 × 2 × 2 × 2 × 2 × 2 × 2)

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Rule #1

2 by b, 4 by m, and 3 by n.

(Multiplying “Like” Bases)

If m and n are any non-zero whole numbers and if b denotes any base,

then bm × bn = bm+n

The above result can be stated more generally if we replace…

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nextLet’s look at a typical question that

we might ask a student to answer…

For what value of x is it true that 35 × 36 = 3x?

This is an application of Rule #1.

b = 3, m = 5 and n = 6.

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In other words… 35 × 36 = 35+6 = 311.

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Notice that the answer is x = 11, not

x = 311. We worded the question the way we did in order to emphasize the role the

exponents played.

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Notes

Of course, if you wanted to, you could rewrite 35 as 243 and 36 as 729, and then multiply 243 by 729 to obtain 177,147,

which is the value of 311. However, that obscures how convenient it is to use the

arithmetic of exponents.

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The point is that if you didn’t know Rule #1 but you knew the definition of 35 and 36, you could have derived the rule just by “returning to the basics”.

Notes

That is…next


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35 36×3 × 3 × 3 × 3 × 3

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= 3x

3 × 3 × 3 × 3 × 3 × 3

Stated verbally, the product of five factors of 3 multiplied by the product of

six factors of 3 gives us the product of eleven factors of 3.

1 2 3 4 5 6 7 8 9 10 1111

×

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Notice that Rule #1 applied to the situation when the bases were the same

but the exponents were different.

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Warning about Blind Memorization

This should not be confused with the case in which the exponents are the

same but the bases are different.

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To see if students understand this subtlety, you might want them to attempt to answer

the following question…

For what value of x is it true that 34 × 24 = 6x?


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If they have memorized Rule #1 without understanding it (such as in the form

“when we multiply, we add the exponents”), they are likely to give the answer x = 8; rather than the correct

answer, which is x = 4.


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If we have them return to basics and use the definitions correctly, they will see that

34 = 3 × 3 × 3 × 3 and 24 = 2 × 2 × 2 × 2.

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Hence…

(3 × 3 × 3 × 3) (2 × 2 × 2 × 2) =×

(3 × 2) (3 × 2) (3 × 2) (3 × 2) =× × ×

(3 × 2)4 = 64

34 × 24 =

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Rule #1a

In the above discussion, if we replace 3 by b, 2 by c and 4 by n, we get the more

general rule…

(Multiplying “Like” Exponents)

If b and c are any numbers and n is any positive whole number,

then bn × cn = (b × c)n

In other words, when we multiply “like exponents”, we multiply the bases and

keep the common exponent.

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At any rate, returning to our main theme, let’s see what happens when we divide

“like” bases. In terms of taking a guess, we know that division is the inverse of

multiplication and that subtraction is the inverse of addition.

Therefore, since we add exponents when we multiply like bases, it would seem

that when we divide like bases we should subtract the exponents.

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next Let’s see if our intuition is correct by

doing a division problem using the basic definition of a non-zero whole number

exponent. To this end, let’s see how we might answer the question below.

For what value of x is it true that 26 ÷ 22 = 2x?

(2 × 2 × 2 × 2 × 2 × 2) ÷ (2 × 2) = 2x

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Using the basic definition we may rewrite 26 ÷ 22 as…

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Since the quotient of two numbers remains unchanged if each term is divided

by the same (non zero) number, we may cancel two factors of 2 from both the dividend and the divisor to obtain…

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(2 × 2 × 2 × 2 × 2 × 2) ÷ (2 × 2) =

2 × 2 × 2 × 2 × 2 × 2 2 × 2

… leaving us with four factors of 2 in the numerator which equals 24.

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2 × 2 × 2 × 2

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Rule #2

(Dividing “Like” Exponents)

If m and n are any non-zero whole numbers and if b denotes any base, then

bm ÷ bn = b m–n

The key point is that when we divided 26 by 22, we subtracted the exponents.

We did not divide them!

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This result can be stated more generally if we replace 2 by b, 6 by m and 2 by n.

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Historical Notenext

Before the invention of the calculator, it was often cumbersome to multiply and divide numbers. The Scottish mathematician, John Napier (1550 - 1617) invented logarithms (in

effect, another name for exponents).

What Rules #1 and #2 tell us is that if we work with exponents, multiplication problems can be replaced by equivalent addition problems

and division problems can be replaced by equivalent subtraction problems.

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Historical Notenext

In this sense, since it is usually easier to add than to multiply and to subtract than to divide,

the use of logarithms became a helpful computational tool. Later, the slide rule was invented and this served as a portable table

of logarithms.

Today, the study of exponents and logarithms still remains important, but not for the

purpose of simplifying computations. Indeed, the calculator does this task much more

quickly and much more accurately.

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However, a reasonable question to ask is “Is there ever a time when it is correct to multiply the two exponents?” The fact that there is a computational situation in which we multiply the exponents can be

seen when we answer the following question…

For what value of x is it true that (24)3 = 2x?

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To find the answer, let’s once again return to the basic definition of an exponent.

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Since everything in parentheses is treated as a single number, ( )3 means

( ) × ( ) × ( ). Hence, (24)3 means 24 × 24 × 24 which is the product of four

factors of 2, multiplied by the product of four more factors of 2, multiplied by four more factors of 2, or altogether, it’s the

product of 12 factors of 2.

In terms of the basic definition,

(24)3 means 24 × 24 × 24, which in turn means…

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24 24×2 × 2 × 2 × 2

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1 2 3 4 5 6 7 8

×

× 24

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

9 10 11 12

(24)3 = 24 × 24 × 24, = 212

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Rule #3

Again, the above result can be stated more generally if we replace 2 by b,

4 by m and 3 by n. The resulting statement is then the general result…

(Raising a Power to a Power)

If m and n are any non-zero whole numbers and if b denotes any base, then

(bm)n = bmn.

In other words, to raise a power to a power, we multiply the exponents.

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We numbered our rules rather arbitrarily, so let’s just summarize

what we have done without referring to a rule by number.

Keep in mind that if you don’t remember the

rule, you can always re-derive it by going back to the basic definitions.

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To multiply two numbers that have the same base, we keep the common base

and add the two exponents.

Example…

38 × 35 = 38 + 5 = 313

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To divide two numbers that have the same base, we keep the common base and

subtract the two exponents.

Example…

38 ÷ 35 = 38 – 5 = 33

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To multiply two numbers that have the same exponents, we keep the common

exponent and add the two bases.

Example…

38 × 48 = (3 × 4)8

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To raise the power of a base to a power, we multiply the two exponents but leave

the base as is.

Example…

(38)5 = 38×5 = 340

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In teaching students the arithmetic of exponents, do not have them memorize the rules. Instead have them work through the rules by seeing what happens when they

apply the basic definitions. Our experiences shows that once students have internalized why the rules are as they are, they almost automatically become better at doing the

computations correctly.

Key Point

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In the next presentation, we will begin the more general discussion of

integer exponents.


Integer Exponents

Arithmetic of Positive Integer Exponents © Math As A Second Language All Rights Reserved next #10...

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