Arithmetic of Positive Integer Exponents © Math As A Second Language All Rights Reserved next #10 Taking the Fear out of Math 2 8 × 2 4

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… next © Math As A Second Language All Rights Reserved 3 / / 7 might tempt students to conclude that… 3 / / 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 © 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… = 4 7

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 2 4 means 2 × 2 × 2 × 2 or 16, and that 2 3 means 2 × 2 × 2 or 8.

Hence… © Math As A Second Language All Rights Reserved next × 2 × 22 × 2 × 2 × …and 24 is a great deal less than 4 7 (which is 4 × 4 × 4 × 4 × 4 × 4 × 4 or 16,384). + next = 24

The key point to observe in the expression 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 × 2 × 2 × 22 × 2 × 2 × 2 816× next = 2 7 ×2 × 2 × 2 × 22 × 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 2 7. (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 b m × b n = b m+n The above result can be stated more generally if we replace…

next © Math As A Second Language All Rights Reserved next Let’s look at a typical question that we might ask a student to answer… For what value of x is it true that 3 5 × 3 6 = 3 x ? This is an application of Rule #1. b = 3, m = 5 and n = 6. next In other words… 3 5 × 3 6 = = 3 11.

© Math As A Second Language All Rights Reserved Notice that the answer is x = 11, not x = 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 3 5 as 243 and 3 6 as 729, and then multiply 243 by 729 to obtain 177,147, which is the value of 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 3 5 and 3 6, 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 × 3 × 3 × 3 × 3 × 3 next = 3 x 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 ×

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 3 4 × 2 4 = 6 x ? 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 3 4 = 3 × 3 × 3 × 3 and 2 4 = 2 × 2 × 2 × 2. next Hence… (3 × 3 × 3 × 3)(2 × 2 × 2 × 2) =× (3 × 2) (3 × 2) =××× (3 × 2) 4 = × 2 4 =

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 b n × c n = (b × c) n In other words, when we multiply “like exponents”, we multiply the bases and keep the common exponent. 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

© 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 2 6 ÷ 2 2 = 2 x ? (2 × 2 × 2 × 2 × 2 × 2) ÷ (2 × 2) = 2 x next Using the basic definition we may rewrite 2 6 ÷ 2 2 as…

© 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 2 4. next 2 × 2 × 2 × 22 × 2 × 2 × 2

© 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 b m ÷ b n = b m–n The key point is that when we divided 2 6 by 2 2, 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.

© Math As A Second Language All Rights Reserved Historical Note next Before the invention of the calculator, it was often cumbersome to multiply and divide numbers. The Scottish mathematician, John Napier ( ) 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 Note next 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 (2 4 ) 3 = 2 x ?

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, (2 4 ) 3 means 2 4 × 2 4 × 2 4 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, (2 4 ) 3 means 2 4 × 2 4 × 2 4, which in turn means… © Math As A Second Language All Rights Reserved next × 2 × 2 × 2 × 2 next × × × 2 × 2 × 2× (2 4 ) 3 = 2 4 × 2 4 × 2 4, = 2 12

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 (b m ) n = b mn. 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.

© 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… 3 8 × 3 5 = = 3 13

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… 3 8 ÷ 3 5 = 3 8 – 5 = 3 3

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… 3 8 × 4 8 = (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… (3 8 ) 5 = 3 8×5 = 3 40

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