The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 7 Part 1

The Arithmetic of Exponents The Arithmetic of Exponents When the exponents are positive whole numbers! © 2007 Herbert I. Gross next

Sometimes the symbolism we use in doing mathematics goads us into making mistakes. In other words, if we don’t understand what the concept is, the symbols that denote the concept can trick us into doing the wrong thing. © 2007 Herbert I. Gross next

When beginning students are asked to compute sums like , the + sign might tempt them into answering… © 2007 Herbert I. Gross next Be Careful! = 4 7 That is, they might add the two bases (2 + 2 = 4 ) and/or the two exponents (4 + 3 = 7).

Thinking --- keep the common denominator and add the two numerators. In other words, they might tend to keep the common base (as if it was the common denominator) and add the two exponents (as if they were the numerators). In this way, they would write = 2 7 next …or they might confuse adding 2 3 and 2 4 with the rule for adding fractions.

Knowing the correct meaning (definition) is clearly crucial if students are to avoid making errors such as these. For example, once students refer to the basic definition of an exponent, they will see that… © 2007 Herbert I. Gross next 2 4 = 2 × 2 × 2 × 2 = = 2 × 2 × 2 = = + 24 Clearly, 24 is much less than either 4 7 (which is 16,384) or 2 7 (which is 128). next

In the expression observe that the group of four factors of 2 and the group of three factors of 2 are separated by a plus sign. Key Point Thus, we are NOT multiplying seven factors of 2.

If the expression had been 2 4 × 2 3 instead of , then we would have had seven factors of 2. © 2007 Herbert I. Gross next × 2 × 2 × 2 × × 2 × 2 × = 2 7 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.

2 4 × 2 3 = This result can be stated more generally if we replace 2 by b, 2 4 × 2 3 = © 2007 Herbert I. Gross next Rule 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 This now constitutes the more general result, which we will call Rule 1… next b b b 4 by m, and 3 by n. m m n n

For what value of x is it true that 3 5 × 3 6 = 3 x ? Practice Problem 1 © 2007 Herbert I. Gross Answer: x = 11 Solution: This is an application of Rule 1 with b = 3, m = 5 and n = 6. In other words, by Rule #1, 3 5 × 3 6 = = next

Notice that the answer is x = 11, not x = We worded the question the way we did in order to emphasize the point that when multiplying like bases we add the exponents. The problem doesn’t require us to rewrite 3 11 in place value notation © 2007 Herbert I. Gross next Note next (The answer would have been 3 11 if the question had been “For what value of x is it true that 3 5 × 3 6 = x?”)

© 2007 Herbert I. Gross next BASICS next 3535 × × = × 3 × 3 × 3 × × 3 × 3 × 3 × 3 × Specifically, the product of five factors of 3 If you didn’t know Rule 1, but you knew the definition of 3 5 and 3 6, you could still obtain the correct answer just by returning to the basics. multiplied by the product of six factors of 3 gives us the product of eleven factors of 3. next

The trouble with memorizing without understanding is that things that are quite different might seem to be the same. To apply this warning to our present discussion, notice that Rule 1 applied to the situation when… the bases were the same, but the exponents were different. © 2007 Herbert I. Gross next A Warning About Blind Memorization

© 2007 Herbert I. Gross next A Warning About Blind Memorization This should not be confused with the case in which the exponents are the same, but the bases are different. To see if you understand this subtlety, do the following problem… next

For what value of x is it true that 3 4 × 2 4 = 6 x ? Practice Problem 2 © 2007 Herbert I. Gross Answer: x = 4 Solution: Let's “return to basics”. That is, by definition… next × 3 × 3 × × 2 × 2 × 2

© 2007 Herbert I. Gross Solution: Hence, there are four factors of 3 and four factors of 2… next × 3 × 3 × × 2 × 2 × 2 × ×

which, in turn, may be rewritten as… (3 × 2) © 2007 Herbert I. Gross Solution: But since a product is independent of how the factors are grouped and/or arranged, we may rewrite them as… ( × )× ×× next ( 3 × 3 × 3 × 3 ) × ( 2 × 2 × 2 × 2 ) next

3 4 × 2 4 = (3 × 2) 4 If in the above expression we replace 3 by b, 2 by c, and 4 by n, we get the general rule... © 2007 Herbert I. Gross next 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 That is, when we multiply like exponents, we multiply the bases and keep the common exponent. next

The point is that the rule for what we do when the bases are the same is quite different from the rule for what we do when the exponents are the same. However, if we understand the basic principles, we will never confuse these two rules. © 2007 Herbert I. Gross next At any rate, returning now to our main theme… let's see what happens when we divide like bases. To this end, try answering the following question.

For what value of x is it true that 2 6 ÷ 2 2 = 2 x ? Practice Problem 3 © 2007 Herbert I. Gross Answer: x = 4 Solution: Using the basic definition, we may write… next 2 2 × × 2 × 2 × 2 × 2 × 2 ÷ ÷

© 2007 Herbert I. Gross Solution: 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 ) leaving four factors of 2. next 2 × 2 × 2 × 2

The key point is that when we divided 2 6 by 2 2, we subtracted the exponents. © 2007 Herbert I. Gross next Key Point We did not divide them! The above result can be stated more generally if again we replace 2 by b, 6 by m and 2 by n. next

The resulting statement is then the general result… © 2007 Herbert I. Gross Rule Dividing Like Bases If m and n are any non-zero whole numbers such that m > n and if b denotes any base, then b m ÷ b n = b m – n. (The case in which m ≤ n will be discussed later in part 2 of this lesson.) next

Historical Note 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. © 2006 Herbert I. Gross © 2007 Herbert I. Gross next

Historical Note In this sense, since it is usually easier to add than to multiply and to subtract rather 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. © 2006 Herbert I. Gross © 2007 Herbert I. Gross next

Today, the study of exponents and logarithms still remains important; but not for the purpose of simplifying computations. © 2007 Herbert I. Gross next Indeed, the calculator does this task much more quickly and much more accurately.

© 2007 Herbert I. Gross next While the use of exponents allows us to replace multiplication problems by addition problems, there are times when we do multiply exponents. To get an idea of when we do this, try the following Practice Question. next

For what value of x is it true that (2 4 ) 3 = 2 x ? Practice Problem 4 © 2007 Herbert I. Gross Answer: x = 12 Solution: Since everything in parentheses is treated as a single number, ( ) 3 means… next ( ) × ( ) × ( ) Hence, (2 4 ) 3 means 2 4 × 2 4 × 2 4

© 2007 Herbert I. Gross Solution: next (2 4 ) 3 Hence, counting the factors of 2… (2 4 ) 3 = 2 x 12 next is, by definition, the product of four factors of 2 multiplied by the product of four more factors of 2, multiplied by four more factors of 2. That is… (2 4 ) 3 2 × 2 × 2 × 2 × × = 2 4 × 2 4 × 2 4 next

© 2007 Herbert I. Gross next Note on Notation Notice that can be interpreted in two different ways For example, as we did in this problem, it may be read as (2 4 ) 3 which, as we’ve just seen, is the same as However, it may also be read as if it were. 2 (4 ) = To avoid this confusion, we shall agree that when the parentheses are omitted we shall read and accordingly, when we want to multiply the two exponents, we will use parentheses and write (2 4 ) 3. 2 (4 ) as 3 Since 4 3 = 4 × 4 × 4 or 64, it means next 2 (4 ) 3

The result (2 4 ) 3 = 2 4×3 can be stated more generally by replacing 2 by b, 4 by m, and 3 by n. The resulting statement is then the general result… © 2007 Herbert I. Gross next Rule 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 m×n. That is: to raise a power to a power, we multiply the exponents. next

© 2007 Herbert I. Gross next 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 That is, when we multiply like exponents, we multiply the bases and keep the common exponent. next Rule 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 Summary of Rules

© 2007 Herbert I. Gross next Rule 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 m×n. That is: to raise a power to a power, That is: to raise a power to a power, we multiply the exponents. next Rule Dividing Like Bases If m and n are any non-zero whole numbers such that m > n and if b denotes any base, then b m ÷ b n = b m – n. (The case in which m ≤ n will be discussed later in part 2 of this lesson.) Summary of Rules

© 2007 Herbert I. Gross next This completes Part 1 of our introduction to arithmetic of exponents (that is: the case in which the exponents are positive integers). In part 2 of this lesson, we will discuss how these rules can be extended, so they can be applied even when they are negative integers and/or zero.