Transcript Mathematics as a Second Language

Taking the Fear
out of Math
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#10
28
× 24
Arithmetic of
Positive Integer
Exponents
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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…
3/
2/
+
7
7
might tempt students to conclude that…
3/
2/ = 5/
+
7
7
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.
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…
4
2
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+
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3
2 =
7
4
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The proper thing to do, especially
when you are in doubt, is to return to
the basic definitions.
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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Hence…
4
2
+
2×2×2×2
16
3
2
2×2×2
+
8
= 24
…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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Key Point
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.
Thus, we are not multiplying
seven factors of 2.
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However, had there been a times sign
then we would have had seven factors
of 2. In other words…
4
2
×
2×2×2×2
×
16
×
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3
2
2×2×2
7
8 == 128
2
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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) =
(2 × 2 × 2 × 2 × 2 × 2 × 2)
… leaving us with seven factors of 2
which equals 27.
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The above result can be stated more
generally if we replace…
2 by b, 4 by m, and 3 by n.
Rule #1
(Multiplying “Like” Bases)
If m and n are any non-zero whole
numbers and if b denotes any base,
then bm × bn = bm+n
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Let’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.
In other words… 35 × 36 = 35+6 = 311.
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Notes
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.
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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Notes
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”.
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That is…
×
5
3
6
3
=
x
3
3 × 3 × 3 × 3 × 3 ×3 × 3 × 3 × 3 × 3 × 3
1
2
3
4
5
6
7
8
9 10 11
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.
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Warning about
Blind Memorization
Notice that Rule #1 applied to the
situation when the bases were the same
but the exponents were different.
This should not be confused with the
case in which the exponents are the
same but the bases are different.
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Warning about
Blind Memorization
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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Warning about
Blind Memorization
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.
Hence…
34 × 24 =
(3 × 3 × 3 × 3)
× (2 × 2 × 2 × 2) =
(3 × 2) × (3 × 2) × (3 × 2) × (3 × 2) =
(3 × 2)4 = 64
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In the above discussion, if we replace
3 by b, 2 by c and 4 by n, we get the more
general rule…
Rule #1a
(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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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?
Using the basic definition we may rewrite
26 ÷ 22 as…
(2 × 2 × 2 × 2 × 2 × 2) ÷ (2 × 2) = 2x
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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…
(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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The key point is that when we divided 26
by 22, we subtracted the exponents.
We did not divide them!
This result can be stated more generally if
we replace 2 by b, 6 by m and 2 by n.
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
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Historical Note
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 Note
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.
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.
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In terms of the basic definition,
(24)3 means 24 × 24 × 24, which in turn
means…
×
4
2
×
4
2
4
2
2×2×2×2 ×2×2×2×2 × 2×2×2×2
1
2
3
4
5
6
7
8
9 10 11 12
(24)3 = 24 × 24 × 24, = 212
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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…
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Rule #3
(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 rederive 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…
8
3
×
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5
=
8
+
5
3 3
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=
13
3
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To divide two numbers that have the same
base, we keep the common base and
subtract the two exponents.
Example…
8
3
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÷
5
=
8
–
5
3 3
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=
3
3
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To multiply two numbers that have the
same exponents, we keep the common
exponent and add the two bases.
Example…
8
3
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×
8
=
4 (3
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×
8
4)
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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…
8
5
=
8×5
(3 ) 3
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=
40
3
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Key Point
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.
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Integer Exponents
In the next presentation,
we will begin the more
general discussion of
integer exponents.
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