Section 4.1 Rules of Exponents


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1 Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells us how many factors of x there are in the expression. For example, is abbreviated as 3 5 ; this means that there are five factors of 3. In this abbreviated form, 3 5 is called the exponential form; 5 is called the exponent and the repeated factor, 3, is called the base. Of course, 3 5 actually has a numerical value; This can be found by using repeated multiplication: ; this value, 243, is called the 5 th power of 3. Sometimes we use the word power in place of the word exponent. Where 3 5 is called the exponential form, is called the expanded form. One of the most important things to know about exponents is this: Exponents have more meaning than they have value. As has been mentioned, the exponent is an abbreviation for repeated multiplication, and x n means n factors of x. To understand this idea better, it might be best to represent exponents (at least for now) in words instead of as numbers. For example, instead of writing 2 3, we could write 2 three. It means, of course, three factors of 2. Writing the exponential form like this will help us to realize that the value of an exponent is not as important as its meaning. So, 2 three means three factors of 2; and 2 four means four factors of 2. Writing them this way should cause one to think more of the meaning of the exponent and not of the value of the exponent. Should we, then, ignore the value of the exponent? No, but we do need to think of its meaning first. We might see 2 3, which is , but our thinking needs to be 2 three. This new way of writing exponents will be used throughout this section as we develop the rules of exponents. Though the meaning of the exponent will be emphasized, we will not ignore its value. Rules of Exponents page 4.11
2 THE FIRST RULES OF EXPONENTS Example 1: For each of the following, state its meaning, write it in expanded form and then evaluate the expression. a) 2 5 b) ( 5) 3 c) 1 8 d) 0 6 Answer: The exponent indicates the number of factors that are to be multiplied. (Some of the multiplication will need to be done off to the side as scratch work.) a) 2 5 means five factors of b) ( 5) 3 means three factors of  5 ( 5) ( 5) ( 5) c) d) Notice, in particular, problems (c) and (d) of the Example 1: (d) and (e) Neither of these should be a surprise. In fact, there is a rule about these very ideas: Exponent Rules for 0 and 1: If n is a positive number, then a) 0 n 0: 0 raised to any positive exponent will always be 0. b) 1 n 1: 1 raised to any positive exponent will always be 1. Exercise 1: For each of the following, state its meaning, write it in expanded form and then evaluate the expression a) 4 3 means b) ( 12) 2 means c) 3 4 means d) 1 6 means e) 0 4 means Rules of Exponents page 4.12
3 In keeping with that most basic understanding of exponents, we can introduce a few rules. The First Power Rule: For any real number x, x 1 x. in other words, x one means one factor of x This can be seen in the prime factorization of a number such as 45: Here we can say that 45 has two factors of 3 and one factor of 5. For this reason, the prime factorization could also be written as : two factors of 3 and one factor of 5. Similarly, we could write the prime factorization of as : one factor of 2 and three factors of 5. Example 2: Answer: Evaluate (or simplify) the expression. a) 2 1 b) ( 5) 1 c) ( 4 5 ) 1 d) x 1 Use the First Power Rule: a) b) ( 5) 15 c) ( 4 5 ) d) x1 x Exercise 2 Evaluate (or simplify) the expression a) 4 1 b) ( 12) 1 c) y 1 d) ( 2 3 ) 1 THE PRODUCT RULE If 2 three means three factors of 2 and 2 four means four factors of 2, then what is the meaning of 2 three 2 four? This is three factors of 2 times four (more) factors of 2, 2 three 2 four can be expanded to (2 2 2) ( ) 2 7, or 2 seven, seven factors of 2. Look at what happened: 2 three 2 four 2 seven. What is being suggested here? That exponents have more meaning than value. Three factors of 2 times four more factors of 2 results in a total of seven factors of 2. This leads to an important rule of exponents, the product rule: The Product Rule for Exponents: If x is a nonzero base and a and b are positive integer exponents, then xa xb x a + b Rules of Exponents page 4.13
4 The product rule is true only when the base is the same for each factor: when multiplying, just add the exponents. Let s better understand the product rule by taking a look at how we can manipulate the following repeated multiplication. x x x x x x x x x x x 10. Notice that there are ten factors of x (you may want to count them). This could also be written as (x x x x) (x x x x x x) x 10. Notice that there are still 10 factors of x, but they ve been broken up so that we now have ( x 4 ) ( x 6 ) ( x four ) ( x six ) x ten x 10 or just x 4 x 6 x 10. This says that we have four factors of x and six more factors of x for a total of ten factors of x. Let s try it again: (x x x x x x x) (x x x) x 10. There are still ten factors of x, but this time they ve been broken up so that we now have x seven x three x ten. or x 7 x 3 x 10. Again, this time we have seven factors of x and three more factors of x, for a total of ten factors of x. Let s try it again: (x x) (x x x x x x x x) x 10 x two x eight x ten. which is the same as saying x 2 x 8 x 10. One more time: (x x x x x) (x x x x x) x 10 x five x five x ten. which is the same as saying x 5 x 5 x 10. Rules of Exponents page 4.14
5 Now you try it. I ll give you nine factors of y, and you find four different ways to rewrite them as I have, above. Write each with all nine factors split into two groups; then write each in its exponential form. I ll show you one possible situation; you find four others. y y y y y y y y y y 9. My example: (y y) (y y y y y y y) y 9 which is the same as: y two y seven y nine. or: y 2 y 7 y y y y y y y y y y y 9 2. y y y y y y y y y y 9 so, y 9 so, y 9 3. y y y y y y y y y y 9 4. y y y y y y y y y y 9 so, y 9 so, y 9 Here is another way to look at the product rule. We can expand each factor and count the number of repeated factors. Example 3: Rewrite each product and power it in its expanded form; then combine all of the factors and abbreviate the result using an exponential form. a) b 3 b 2 b) y 3 y 4 c) x 5 x Answer: Notice that each factor in each problem has the same base. This is important for the pattern to be recognized. a) b 3 b 2 (b b b) (b b) b 5 ; b three b two b five b) y 3 y 4 (y y y) (y y y y) y 7 ; y three y four y seven c) x 5 x (x x x x x) (x) x 6 ; x five x one x six Rules of Exponents page 4.15
6 Exercise 3 Rewrite each product and power in its expanded form; then combine all of the factors and abbreviate the result using an exponential form. a) x 5 x 4 b) y 2 y 3 c) c 6 c 2 d) w 3 w 3 e) k 1 k 1 f) m m 3 g) x 5 x Now let s apply the product rule directly. The Product Rule for Exponents: If x is a nonzero base and a and b are positive integer exponents, then xa xb x a + b Example 4: Use the product rule for exponents to write each of these as one base with one exponent. a) b 3 b 2 b) y 3 y 4 c) x 5 x Answer: Notice that each factor in each problem has the same base. This is important for the pattern to be recognized. a) b 3 b 2 b b 5 (three factors of b and two more factors of b ) b) y 3 y 4 y y 7 (three factors of y and four more factors of y ) c) x 5 x x x 6 (remember that x x 1 ) Rules of Exponents page 4.16
7 Exercise 4 Use the product rule for exponents to write each of these as one base with one exponent. a) x 5 x 4 b) x 3 x 6 c) y 2 y 3 d) c 6 c 2 e) w 3 w 3 f) k 1 k 4 g) m m 3 h) x 5 x Please note: when working with the product of two exponent expressions with the same base, there are only two ways to combine the two bases into one (but they must first be identical bases): 1) write each in expanded form and count 2) use the product rule for exponents: the number of factors of the base, as in as in x 4 x 3 (x x x x) (x x x) x 7. x 4 x 3 x x 7. It s also very important to note that the product rule for exponents cannot be used if the bases are not the same. For example, x 3 y 5 is nothing more than x 3 y 5. This could be thought of as x three y five, three factors of x and five factors of y, but we don t have eight factors of any one variable. THE ZERO POWER RULE You know that 1 is the identity for multiplication. This means that the product of any number, A, and 1 is always that number: A 1 A. We can use this idea, along with the product rule, to introduce a new rule the zero power rule. Consider the product a 4 1 a 4. Also consider the product a 4 a 0 a a 4. So, a 4 a 0 a 4 and a 4 1 a 4 ; This could only mean that a 0 and 1 are the same; in other words, a 0 1. The Zero Power Rule: For any nonzero real number x, x 0 1. in other words, x zero means no factor of x There is only one exception: 0 0 is undefined. Rules of Exponents page 4.17
8 The Exception: The rule itself is explained above, but the exception needs a brief mention. 0 0 is undefined because it puts two rules in conflict with one another, and that s not allowed in mathematics: The Exponent Rule for 0 would state 0 0 0, but The Zero Power Rule would state 0 0 1! But we can t have it both ways: 0 0 can t equal both 0 and 1. Therefore, we say it can t happen at all; in other words, 0 0 is undefined. Let s put The Zero Power Rule into practice: Example 5: a) b 0 1 b) c) d) ( 2) 0 1 e) ( 3) 0 1 f) x 0 1 Exercise 5 Use the Zero Power Rule to evaluate each. a) w 0 b) 7 0 c) ( 4) 0 d) 14 0 e) y 0 f) 1 0 g) 3 0 h) 0 0 To further understand the zero power rule, that a 0 1, we refer back to a 4 a 0 a 4 and recognize that a 0 doesn t add any more factors to the product. Note that it doesn t make the entire product zero. Example 6: Use the product rule for exponents to write each of these as one base with one exponent. a) b 3 b 0 b) y 0 y 4 c) x 0 x Answer: Notice that each factor in each problem has the same base. This is important for the pattern to be recognized. a) b 3 b 0 b b 3 (three factors of b and no more factors of b ) b) y 0 y 4 y y 4 (no factors of y and four factors of y ) c) x 0 x x x 1 x (remember that x 1 x) Rules of Exponents page 4.18
9 Exercise 6 Use the product rule for exponents to write each of these as one base with one exponent. a) x 5 x 0 b) x 0 x 6 c) y 0 y 2 d) c 6 c 0 e) w 9 w 1 f) k 1 k 4 g) m m 0 h) x 0 x 0 THE QUOTIENT RULE A product is an expression of multiplication; a quotient is an expression of division. Usually, a quotient is expressed as a fraction, and we know that we can simplify fractions by canceling common factors in the numerator and the denominator. For example, you can probably see that can reduce by a factor of 6: Recognizing common factors, especially using prime factorization, is another method that works quite well. Writing the numerator and denominator in its prime factored form allows for easy cancellation of common factors: We can do the same for a quotient such as x5 x 3. If we write it in its expanded form, then we ll be able to see that many of the factors will cancel: Of course, we can cancel only common factors, and because the bases are the same, both x, the numerator and denominator have plenty of common factors to cancel. Another way to show the canceling process is to group the same number of common factors in both the x 5 numerator and denominator. For example, with x 3 we can expand both the numerator and denominator and then group three factors of x in each: x 5 x 3 x x x x x x x x (x x x) x x (x x x) x x x 2. Rules of Exponents page 4.19
10 Example 7: Write both the numerator and the denominator in expanded form; then simplify the fraction by canceling any common factors. a) x 8 x 2 x x x x x x x x x x (x x) x x x x x x (x x) x x x x x x x 6 b) y 7 y 6 y y y y y y y y y y y y y (y y y y y y) y (y y y y y y) y or y1 c) w 4 w 4 w w w w w w w w (w w w w) (w w w w) 1 d) p 5 p 0 p p p p p 1 p p p p p p 5 Exercise 7 Write both the numerator and the denominator in expanded form; then simplify the fraction by canceling any common factors. a) x 5 x 4 b) w 8 w 5 c) c 6 c 1 d) p 4 p 0 e) y 9 y 7 f) a 6 a 6 g) m 8 m 4 h) x 7 x Rules of Exponents page
11 Let s return to the example of simplifying x5 x 3 xfive x three : In this example, the numerator has five factors of x and the denominator has three factors of x. In the canceling process, all three of the denominator factors cancel with three of the numerator factors leaving two factors of x in the numerator. It s as if we are eliminating three factors of x. This process suggests a subtraction: five take away three is two Of course, in this process we are subtracting the exponents, just as we are dividing out the number of common factors. The rule that supports this type of dividing out is called the quotient rule. The Quotient Rule (for Exponents): If x is a nonzero base and a and b are positive integer exponents, then x a x b x a b This works only when the base is the same for each factor: when dividing, just subtract the exponents. Example 8: Use the Quotient Rule to simplify each of these. a) x 8 x 2 b) y 7 y 6 c) w 4 w 4 d) p 5 p 0 Answer: Make sure the bases are the same. Then, subtract the exponents to get the result. a) c) x 8 x 2 x 8 2 x 6 b) w 4 w 4 w 4 4 w 0 1 d) y 7 y 6 y 7 6 y 1 or just y p 5 p 0 p 5 0 p 5 Rules of Exponents page
12 Exercise 8 Use the Quotient Rule to simplify each of these. a) c) e) g) x 5 x 4 b) c 6 c 1 d) y 9 y 7 f) v 9 v 3 h) w 8 w 5 p 4 p 0 a 6 a 6 m 8 m 4 i) x 4 x j) k 8 k 7 THE ZERO POWER RULE, REVISITED We can use the Quotient Rule to further explain why x 0 1. We know, of course, that any number divided A by itself is 1: A 1. This is also true when the number is raised to a power. For example, consider , which is just 1. We can consider this fraction in a variety of ways, but it always reduces to 1: (1) Expanding the numerator and denominator: (2) Evaluating the numerator and denominator: (3) Using the Quotient Rule: (2 2 2) (2 2 2) Of course, (3) shows us again that x 0 1. The zero power rule can be confusing. When seeing it or using it, you must first think and remember that exponents have more meaning then they have value. In other words, think about what the zero power means before deciding what its value will be. Rules of Exponents page
13 THE DISTRIBUTIVE PROPERTIES FOR EXPONENTS To this point we have used the rules of exponents with bases that are integers. However, it s quite possible that the base can be a fraction. For example, consider ( a b ) 3 : when expanded it becomes a b a b a b a a a b b b a3 b 3. Similarly, consider (a b) 3 : when expanded it becomes... The associative property allows us to write these without parentheses the commutative property allows us to switch these around (a b) (a b) (a b) a b a b a b a a a b b b the associative property allows us to regroup them (a a a) (b b b) and the definition of the exponent allows to abbreviate it as a 3 b 3 In general, if a quantity is a product (multiplication) or a quotient (such as a fraction), then we can distribute an exponent to each part. The Distributive Properties for Exponents: a) If x and y are any bases, then (x y) a x a y a b) If x and y are any bases, and y 0, then x y a x a y a The reason the Distributive Properties for Exponents work so well for multiplication is that exponents are compatible with multiplication and division. Consider this: Multiplication is the abbreviation for repeated addition, so multiplication is compatible with the Distributive Property for Addition: a(b + c) a b + a c Likewise, an exponent is the abbreviation for repeated multiplication, so the exponent is compatible with the Distributive Property for Exponents. (x y) a x a y a However, the Distributive Property for Exponents is not compatible with addition; in other words (x + y) a x a + y a Rules of Exponents page
14 Example 9: Use the Distributive Properties for Exponents to rewrite each of these. Simplify if possible. a) (a b) 5 b) (3 w) 4 c) 2 5 p d) 7y 2 w e) ( 2 m) 3 f) ( 5 r) 2 Answer: a) (a b) 5 a 5 b 5 b) (3 w) w 4 81w 4 c) 2 5 p 25 p 5 32 p 5 d) 7y 2 w (7y)2 w 2 72 y2 w 2 49y2 w 2 e) ( 2 m) 3 ( 2) 3 m 38m 3 f) ( 5 r) 2 ( 5) 2 r 2 25r 2 Exercise 9 Use the Distributive Properties for Exponents to rewrite each of these. Simplify if possible. a) (w y) 6 b) (4p) 3 c) ( 2 x) 4 d) ( 10c) 3 e) a 9 b f) 5 3 y g) x 2 6 h) 3y 3 2 i)  11x 9 2 j)  1k 2d 5 k) (2x) 1 l) (10c) 0 m) x 1 9 n) 4 0 d Rules of Exponents page
15 THE POWER RULE OF EXPONENTS The last rule to be presented in this section is the Power Rule (of Exponents): The Power Rule (of Exponents) also known as The Power of a Power : (x a ) b x a b The explanation to this rule is best seen in an example. Consider (x 2 ) 3 ; to best understand this, we should refer back to the definition of the exponent, an abbreviation for repeated multiplication. (x 2 ) 3, or (x 2 ) three, means three factors of x 2 : x 2 x 2 x 2 x x 6. Notice that in x we actually have repeated addition in the exponent, this repeated addition could be abbreviated as x three 2 s x 3 2 (or x 2 3 ) x 6. Notice also that (x 3 ) 2 x 3 2 x 2 3 x 6 ; this suggests that the commutative property can also be applied. Let s look at two numerical examples to see it just one more way; consider both (2 2 ) 3 and (2 3 ) 2 First, recall that So, in applying the order of operations to (2 2 ) 3, we get (4) 3 which is 64; and, in applying the order of operations to (2 3 ) 2, we get (8) 2 which is 64. Lastly, in applying the power rule to (2 2 ) 3, we get which is 64. and to (2 3 ) 2, we get which is 64. Rules of Exponents page
16 Example 10: Use the Power Rule to rewrite each of these. a) (w 4 ) 3 b) (x 3 y) 5 c) a4 p 3 2 Answer: Notice that, in (b) and (c), the power rule can be combined with the distributive rules. (a) (w 4 ) 3 w 4 3 w 12 (b) (x 3 y) 5 (x 3 ) 5 y 5 x 15 y 5 (c) a4 p 3 2 (a 4 ) 2 (p 3 ) 2 a 8 p 6 Exercise 10 Use the Power Rule to rewrite each of these. Simplify if possible. a) (c 5 ) 2 b) (m 4 ) 6 c) (3w 2 ) 4 d) (a 6 b) 3 e) (2y 5 ) 6 f) (c 2 d 5 ) 3 g) n5 2 9 h) 3 c 3 3 i) y w 4 3 j) p5 k 6 4 k) (c 2 d 5 ) 1 l) (c 2 d 5 ) 0 m) y w 4 1 n) p5 k 6 0 Rules of Exponents page
17 Answers to each Exercise Section 4.1 Exercise 1: a) 4 3 means three factors of b) ( 12) 2 means two factors of ( 12) ( 12) ( 12) c) 3 4 means four factors of d) 1 6 means six factors of e) 0 4 means four factors of Exercise 2: a) b) ( 12) 112 c) y 1 y d) ( 2 3 ) Exercise 3: a) x 5 x 4 (x x x x x) (x x x x) x 9 b) y 2 y 3 (y y) (y y y) y 5 c) c 6 c 2 (c c c c c c) (c c) c 8 d) w 3 w 3 (w w w) (w w w) w 6 e) k 1 k 1 (k) (k) k 2 f) m m 3 (m) (m m m) m 4 g) x 5 x (x x x x x) (x) x 6 Exercise 4: a) x 9 b) x 9 c) y 5 d) c 8 e) w 6 f) k 5 g) m 4 h) x 6 Exercise 5: a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 g) 1 h) undefined Exercise 6: a) x 5 b) x 6 c) y 2 d) c 6 e) w 10 f) k 5 g) m 1 m h) x 0 1 Rules of Exponents page
18 Exercise 7: a) b) c) d) e) f) g) h) x 5 x 4 w 8 w 5 c 6 c 1 p 4 p 0 y 9 y 7 a 6 a 6 m 8 m 4 x 7 x 1 x x x x x x x x x (x x x x) x (x x x x) x w w w w w w w w w w w w w c c c c c c c p p p p 1 p 4 y y y y y y y y y y y y y y y y (w w w w w ) w w w (w w w w w) w 3 (c) c c c c c (c) c 5 a a a a a a a a a a a a (a a a a a a) (a a a a a a) 1 m m m m m m m m m m m m x x x x x x x x (y y y y y y y) y y (y y y y y y y) y 2 (m m m m) m m m m (m m m m) m 4 (x) x x x x x x (x) x 6 Exercise 8: a) x 1 or just x b) w 3 c) c 5 d) p 4 e) y 2 f) a 0 1 g) v 6 h) m 4 i) x 3 j) k 1 or just k Exercise 9: a) w 6 y 6 b) 64p 3 c) 16x 4 d)  1,000 c 3 e) i) m) a 9 b 9 f) 125 y 3 g) x 2 36 h) 27y x 2 81 j)  1k 5 32d 5 k) 2x l) 1 x 9 n) 1 Exercise 10: a) c 10 b) m 24 c) 81w 8 d) a 18 b 3 e) 64y 30 f) c 6 d 15 n 10 g) 81 h) 27 c 9 y 3 p 20 i) w 12 j) k 24 k) c 2 d 5 l) 1 y m) w 4 n) 1 Section 4.1 Focus Exercises Rules of Exponents page
19 1. Evaluate each using a rule of exponents. a) x 0 b) 4 1 c) ( 4) 0 d) ( 6) 1 e) y 1 f) g) h) Write each product as one base with one exponent. You may expand and count the number of factors or use the product rule. a) x 2 x 6 b) v 5 v 9 c) y 3 y d) c c 4 e) p 5 p 0 f) k 0 k 6 g) w w 0 h) y 0 y 0 i) x 4 x 10 j) c 3 c 3 k) p 1 p 1 l) m 0 q 4 3. Write each quotient as one base with one exponent. You may expand and count the number of factors or use the quotient rule. a) x 9 x 5 b) w 6 w 3 c) c 5 c 1 d) p 3 p 0 e) y 8 y f) a 7 a 7 g) m 12 m 6 h) x 3 x 2 i) x x j) c 3 c 3 k) y 10 y 2 l) m 21 m 7 Rules of Exponents page
20 4. Use the Distributive Properties for Exponents to rewrite each of these. Simplify if possible. a) ( k h ) 4 b) ( 3m ) 4 c) (  3 p ) 2 d) (  2x ) 3 e) y 5 x f) 5 0 h g) p 1 6 h) 2h 3 3 i)  7p 2 8 j)  1v 3 2w k) ( 2p ) 5 l) ( 8x ) 0 5. Use the Power Rule to rewrite each of these. Simplify if possible. a) ( x 2 ) 4 b) ( r 5 ) 3 c) ( 2k 3 ) 4 d) ( y 4 b 3 ) 2 e) ( 4c 3 ) 3 f) ( xw 2 ) 5 g) p2 m 3 2 h) y x 4 3 i) 2 k 2 3 j) m0 v 1 4 k) ( x 3 w 4 ) 1 l) ( x 6 w ) 0 m) 4c k 9 1 n) m4 v 4 0 Rules of Exponents page
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