arithmetic of positive integer exponents © math as a second language all rights reserved next #10...
TRANSCRIPT
Arithmetic of Positive Integer
Exponents
© Math As A Second Language All Rights Reserved
next
#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…
nextnext
© Math As A Second Language All Rights Reserved
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
next
A similar problem might occur when beginning students are first asked to
compute such sums as 24 + 23.
next
© Math As A Second Language All Rights Reserved
next
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.
next
© Math As A Second Language All Rights Reserved
next
In this case, we know that 24 means
2 × 2 × 2 × 2 or 16, and that 23 means
2 × 2 × 2 or 8.
Hence…next
© Math As A Second Language All Rights Reserved
next
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).
+
nextnextnext
= 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.
next
© Math As A Second Language All Rights Reserved
next
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…
next
© Math As A Second Language All Rights Reserved
next
24 23×2 × 2 × 22 × 2 × 2 × 2
816 ×
nextnextnext
= 27
×2 × 2 × 2 × 2 2 × 2 × 2
= 128
next
© Math As A Second Language All Rights Reserved
next
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)
next
© Math As A Second Language All Rights Reserved
next
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…
next
© Math As A Second Language All Rights Reserved
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.
nextnext
In other words… 35 × 36 = 35+6 = 311.
next
© Math As A Second Language All Rights Reserved
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.
next
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.
next
© Math As A Second Language All Rights Reserved
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
© Math As A Second Language All Rights Reserved
next
35 36×3 × 3 × 3 × 3 × 3
nextnextnext
= 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
×
next
© Math As A Second Language All Rights Reserved
Notice that Rule #1 applied to the situation when the bases were the same
but the exponents were different.
next
Warning about Blind Memorization
This should not be confused with the case in which the exponents are the
same but the bases are different.
next
© Math As A Second Language All Rights Reserved
next
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?
Warning about Blind Memorization
next
© Math As A Second Language All Rights Reserved
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.
Warning about Blind Memorization
next
© Math As A Second Language All Rights Reserved
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.
next
Hence…
(3 × 3 × 3 × 3) (2 × 2 × 2 × 2) =×
(3 × 2) (3 × 2) (3 × 2) (3 × 2) =× × ×
(3 × 2)4 = 64
34 × 24 =
next
© Math As A Second Language All Rights Reserved
next
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.
next
next
© Math As A Second Language All Rights Reserved
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.
next
next
© Math As A Second Language All Rights Reserved
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
next
Using the basic definition we may rewrite 26 ÷ 22 as…
next
© Math As A Second Language All Rights Reserved
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…
next
(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.
nextnext
2 × 2 × 2 × 2
next
© Math As A Second Language All Rights Reserved
next
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!
next
This result can be stated more generally if we replace 2 by b, 6 by m and 2 by n.
next
© Math As A Second Language All Rights Reserved
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.
next
© Math As A Second Language All Rights Reserved
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.
next
© Math As A Second Language All Rights Reserved
next
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?
next
© Math As A Second Language All Rights Reserved
To find the answer, let’s once again return to the basic definition of an exponent.
next
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…
next
© Math As A Second Language All Rights Reserved
next
24 24×2 × 2 × 2 × 2
nextnext
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
next
© Math As A Second Language All Rights Reserved
next
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.
next
© Math As A Second Language All Rights Reserved
next
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.
next
© Math As A Second Language All Rights Reserved
next
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
next
© Math As A Second Language All Rights Reserved
next
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
next
© Math As A Second Language All Rights Reserved
next
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
next
© Math As A Second Language All Rights Reserved
next
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
next
© Math As A Second Language All Rights Reserved
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
next
In the next presentation, we will begin the more general discussion of
integer exponents.
© Math As A Second Language All Rights Reserved
Integer Exponents