Properties of Exponents

Transcription

1 Properties of Eponents Eve Rawley, (EveR) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) Say Thanks to the Authors Click (No sign in required)

2 To access a customizable version of this book, as well as other interactive content, visit AUTHORS Eve Rawley, (EveR) Anne Gloag, (AnneG) Andrew Gloag, (AndrewG) CK- Foundation is a non-profit organization with a mission to reduce the cost of tetbook materials for the K- market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FleBook, CK- intends to pioneer the generation and distribution of high-quality educational content that will serve both as core tet as well as provide an adaptive environment for learning, powered through the FleBook Platform. Copyright 04 CK- Foundation, The names CK- and CK and associated logos and the terms FleBook and FleBook Platform (collectively CK- Marks ) are trademarks and service marks of CK- Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Ecept as otherwise noted, all CK- Content (including CK- Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial.0 Unported (CC BY-NC.0) License ( licenses/by-nc/.0/), as amended and updated by Creative Commons from time to time (the CC License ), which is incorporated herein by this reference. Complete terms can be found at Printed: November 9, 04

3 Chapter. Properties of Eponents CHAPTER Properties of Eponents CHAPTER OUTLINE. Eponent Properties Involving Products. Eponent Properties Involving Quotients. Zero, Negative, and Fractional Eponents

4 .. Eponent Properties Involving Products Eponent Properties Involving Products Learning Objectives Use the product of a power property. Use the power of a product property. Simplify epressions involving product properties of eponents. Introduction Back in chapter, we briefly covered epressions involving eponents, like 5 or. In these epressions, the number on the bottom is called the base and the number on top is the power or eponent. The whole epression is equal to the base multiplied by itself a number of times equal to the eponent; in other words, the eponent tells us how many copies of the base number to multiply together. Eample Write in eponential form. a) b) ( )( )( ) c) y y y y y d) (a)(a)(a)(a) a) because we have factors of b) ( )( )( )( ) because we have factors of (-) c) y y y y y y 5 because we have 5 factors of y d) (a)(a)(a)(a)(a) 4 because we have 4 factors of a When the base is a variable, it s convenient to leave the epression in eponential form; if we didn t write 7, we d have to write instead. But when the base is a number, we can simplify the epression further than that; for eample, 7 equals, but we can multiply all those s to get 8. Let s simplify the epressions from Eample. Eample Simplify. a) b) ( ) c) y 5 d) (a) 4 a) 4

5 Chapter. Properties of Eponents b) ( ) ( )( )( ) 7 c) y 5 is already simplified d) (a) 4 (a)(a)(a)(a) a a a a 8a 4 Be careful when taking powers of negative numbers. Remember these rules: (negative number) (positive number)negative number (negative number) (negative number)positive number So even powers of negative numbers are always positive. Since there are an even number of factors, we pair up the negative numbers and all the negatives cancel out. ( ) 6 ( )( )( )( )( )( )( )( {z ) ( } )( {z ) ( } )( {z ) +64 } And odd powers of negative numbers are always negative. Since there are an odd number of factors, we can still pair up negative numbers to get positive numbers, but there will always be one negative factor left over, so the answer is negative: ( ) 5 ( )( )( )( )( )( )( {z ) ( } )( {z ) ( ) } {z } Use the Product of Powers Property So what happens when we multiply one power of by another? Let s see what happens when we multiply to the power of 5 by cubed. To illustrate better, we ll use the full factored form for each: ( ) ( ) ( ) {z } {z } {z } 5 8 So 5 8. You may already see the pattern to multiplying powers, but let s confirm it with another eample. We ll multiply squared by to the power of 4: ( ) ( ) ( ) {z } {z } {z } 4 So 4 6. Look carefully at the powers and how many factors there are in each calculation. 5 s times s equals (5 + )8 s. s times 4 s equals ( + 4)6 s. You should see that when we take the product of two powers of, the number of s in the answer is the total number of s in all the terms you are multiplying. In other words, the eponent in the answer is the sum of the eponents in the product. Product Rule for Eponents: n m (n+m) There are some easy mistakes you can make with this rule, however. Let s see how to avoid them. Eample 6

6 .. Eponent Properties Involving Products Multiply. 5 Note that when you use the product rule you don t multiply the bases. In other words, you must avoid the common error of writing 4 5. You can see this is true if you multiply out each epression: 4 times 8 is definitely, not 04. Eample 4 Multiply In this case, we can t actually use the product rule at all, because it only applies to terms that have the same base. In a case like this, where the bases are different, we just have to multiply out the numbers by hand the answer is not 5 or 5 or 6 5 or anything simple like that. Use the Power of a Product Property What happens when we raise a whole epression to a power? Let s take to the power of 4 and cube it. Again we ll use the full factored form for each epression: ( 4 ) f actors o f { to the power 4} ( ) ( ) ( ) So ( 4 ). You can see that when we raise a power of to a new power, the powers multiply. Power Rule for Eponents: ( n ) m (n m) If we have a product of more than one term inside the parentheses, then we have to distribute the eponent over all the factors, like distributing multiplication over addition. For eample: ( y) 4 ( ) 4 (y) 4 8 y 4. Or, writing it out the long way: ( y) 4 ( y)( y)( y)( y)( y)( y)( y)( y) y y y y 8 y 4 Note that this does NOT work if you have a sum or difference inside the parentheses! For eample, (+y) 6 +y. This is an easy mistake to make, but you can avoid it if you remember what an eponent means: if you multiply out (+y) it becomes (+y)(+y), and that s not the same as +y. We ll learn how we can simplify this epression in a later chapter. The following video from YourTeacher.com may make it clearer how the power rule works for a variety of eponential epressions: 4

7 Chapter. Properties of Eponents Eample 5 Simplify the following epressions. a) 5 7 b) 6 c) (4 ) When we re just working with numbers instead of variables, we can use the product rule and the power rule, or we can just do the multiplication and then simplify. a) We can use the product rule first and then evaluate the result: OR we can evaluate each part separately and then multiply them: b) We can use the product rule first and then evaluate the result: OR we can evaluate each part separately and then multiply them: c) We can use the power rule first and then evaluate the result: (4 ) OR we can evaluate the epression inside the parentheses first, and then apply the eponent outside the parentheses: (4 ) (6) Eample 6 Simplify the following epressions. a) 7 b) (y ) 5 When we re just working with variables, all we can do is simplify as much as possible using the product and power rules. a) b) (y ) 5 y 5 y 5 Eample 7 Simplify the following epressions. a) ( y ) (4y ) b) (4yz) ( y ) (yz 4 ) c) (a b ) When we have a mi of numbers and variables, we apply the rules to each number and variable separately. a) First we group like terms together: ( y ) (4y )( 4) ( ) (y y ) Then we multiply the numbers or apply the product rule on each grouping: y 5 b) Group like terms together: (4yz) ( y ) (yz 4 )(4 ) ( ) (y y y) (z z 4 ) Multiply the numbers or apply the product rule on each grouping: 8 y 5 z 5 c) Apply the power rule for each separate term in the parentheses: (a b ) (a ) (b ) Multiply the numbers or apply the power rule for each term 4a 6 b 6 5

8 .. Eponent Properties Involving Products Eample 8 Simplify the following epressions. a) ( ) b) ( y) (y ) c) (4a b ) (ab 4 ) In problems where we need to apply the product and power rules together, we must keep in mind the order of operations. Eponent operations take precedence over multiplication. a) We apply the power rule first: ( ) 4 Then apply the product rule to combine the two terms: 4 7 b) Apply the power rule first: ( y) (y ) ( y) (7 y 6 ) Then apply the product rule to combine the two terms: ( y) (7 y 6 )54 5 y 7 c) Apply the power rule on each of the terms separately: (4a b ) (ab 4 ) (6a 4 b 6 ) (8a b ) Then apply the product rule to combine the two terms: (6a 4 b 6 ) (8a b )8a 7 b 8 Homework Problems Write in eponential notation: ( a)( a)( a)( a) y y y y 5. y y Find each number ( ) 6 8. (0.) 5 9. ( 0.6) 0. (.) + 5. (0.) Multiply and simplify: ( y 4 )( y) 7. (4a )( a)( 5a 4 ) Simplify: 6 8. (a ) 4

9 Chapter. Properties of Eponents 9. (y) 0. (a b ) 4. ( y 4 z ) 5. ( 8) (5). (4a )( a ) 4 4. (y)(y) 5. (y )( y) ( y ) 7

10 .. Eponent Properties Involving Quotients Eponent Properties Involving Quotients Learning Objectives Use the quotient of powers property. Use the power of a quotient property. Simplify epressions involving quotient properties of eponents. Use the Quotient of Powers Property The rules for simplifying quotients of eponents are a lot like the rules for simplifying products. Let s look at what happens when we divide 7 by 4 : 7 4 You can see that when we divide two powers of, the number of s in the solution is the number of s in the top of the fraction minus the number of s in the bottom. In other words, when dividing epressions with the same base, we keep the same base and simply subtract the eponent in the denominator from the eponent in the numerator. Quotient Rule for Eponents: n (n m) m When we have epressions with more than one base, we apply the quotient rule separately for each base: 5 y y y y y y y y y OR 5 y y 5 y y Eample Simplify each of the following epressions using the quotient rule. a) 0 5 b) a6 a c) a5 b 4 a b a) b) a6 a a6 a 5 c) a5 b 4 a b a 5 b 4 a b Now let s see what happens if the eponent in the denominator is bigger than the eponent in the numerator. For eample, what happens when we apply the quotient rule to 4 7? The quotient rule tells us to subtract the eponents. 4 minus 7 is -, so our answer is. A negative eponent! What does that mean? 8

11 Chapter. Properties of Eponents Well, let s look at what we get when we do the division longhand by writing each term in factored form: 4 7 Even when the eponent in the denominator is bigger than the eponent in the numerator, we can still subtract the powers. The s that are left over after the others have been canceled out just end up in the denominator instead of the numerator. Just as 7 would be equal to 4 (or simply ), 4 is equal to. And you can also see that is equal 7 to. We ll learn more about negative eponents shortly. Eample Simplify the following epressions, leaving all eponents positive. a) 6 b) a b 6 a 5 b a) Subtract the eponent in the numerator from the eponent in the denominator and leave the s in the denominator: b) Apply the rule to each variable separately: a b 6 a 5 b b6 a 5 b5 a The Power of a Quotient Property When we raise a whole quotient to a power, another special rule applies. Here is an eample: 4 y y y y y ( ) ( ) ( ) ( ) (y y) (y y) (y y) (y y) y 8 Notice that the eponent outside the parentheses is multiplied by the eponent in the numerator and the eponent in the denominator, separately. This is called the power of a quotient rule: Power Rule for Quotients: n p y n p m y m p Let s apply these new rules to a few eamples. Eample Simplify the following epressions. a) 45 4 b) c) 4 5 Since there are just numbers and no variables, we can evaluate the epressions and get rid of the eponents completely. a) We can use the quotient rule first and then evaluate the result: OR we can evaluate each part separately and then divide:

12 .. Eponent Properties Involving Quotients b) Use the quotient rule first and hen evaluate the result: OR evaluate each part separately and then reduce: Notice that it makes more sense to apply the quotient rule first for eamples (a) and (b). Applying the eponent rules to simplify the epression before plugging in actual numbers means that we end up with smaller, easier numbers to work with. c) Use the power rule for quotients first and then evaluate the result: OR evaluate inside the parentheses first and then apply the eponent: Eample 4 Simplify the following epressions: a) 5 b) 4 5 a) Use the quotient rule: b) Use the power rule for quotients and then the quotient rule: OR use the quotient rule inside the parentheses first, then apply the power rule: Eample 5 Simplify the following epressions. a) 6 y y a b b) 8a 7 b 4 5 ( ) 5 5 When we have a mi of numbers and variables, we apply the rules to each number or each variable separately. a) Group like terms together: 6 y y 6 y y Then reduce the numbers and apply the quotient rule on each fraction to get y. b) Apply the quotient rule inside the parentheses first: a b 8a 7 b b 4a 4 Then apply the power rule for quotients: Eample 6 Simplify the following epressions. b 4a 4 b 4 6a 8 a) ( ) 6 4 b) 6a 4b b 5 a 6 In problems where we need to apply several rules together, we must keep the order of operations in mind. a) We apply the power rule first on the first term: 0

13 Chapter. Properties of Eponents Then apply the quotient rule to simplify the fraction: ( ) And finally simplify with the product rule: b) 6a 4b 5 b a 6 Simplify inside the parentheses by reducing the numbers: 4a b b 5 a 6 Then apply the power rule to the first fraction: 4a b 5 b 64a6 a6 b 5 b a 6 Group like terms together: 64a 6 b 5 b a6 64 a6 a b 6 b 5 And apply the quotient rule to each fraction: 64 a6 a 6 b 64 b5 a 0 b Homework Problems Evaluate the following epressions

14 .. Eponent Properties Involving Quotients Simplify the following epressions. a 9. a a b 4 a b 6 y y 5. 6a a a 4 5a 0 4 5y 6 0y 5 6 y 4 y 4 6a 4b 4 5b a (ab) (4a b 4 ) (6a b) 4 (a bc )(6abc ) 4ab c (a bc )(6abc ) y z z 4ab c for a,b, and c for,y, and z. y y for,y 4. y y for 0,y 6 5. If a and b, simplify (a b)(bc) a c as much as possible.

15 Chapter. Properties of Eponents. Zero, Negative, and Fractional Eponents Learning Objectives Simplify epressions with zero eponents. Simplify epressions with negative eponents. Simplify epression with fractional eponents. Evaluate eponential epressions. Introduction There are many interesting concepts that arise when contemplating the product and quotient rule for eponents. You may have already been wondering about different values for the eponents. For eample, so far we have only considered positive, whole numbers for the eponent. So called natural numbers (or counting numbers) are easy to consider, but even with the everyday things around us we think about questions such as is it possible to have a negative amount of money? or what would one and a half pairs of shoes look like? In this lesson, we consider what happens when the eponent is not a natural number. We will start with What happens when the eponent is zero? Simplify Epressions with Eponents of Zero Let us look again at the quotient rule for eponents (that n n m take the eample of 4 divided by 4. m ) and consider what happens when n m. Let s 4 4 (4 4) 0 Now we arrived at the quotient rule by considering how the factors of cancel in such a fraction. Let s do that again with our eample of 4 divided by 4. So 0. This works for any value of the eponent, not just n n n n 0 Since there is the same number of factors in the numerator as in the denominator, they cancel each other out and we obtain 0. The zero eponent rule says that any number raised to the power zero is one. Zero Rule for Eponents: 0, 6 0

16 .. Zero, Negative, and Fractional Eponents Simplify Epressions With Negative Eponents Again we will look at the quotient rule for eponents (that n n m m > n. Let s take the eample of 4 divided by 6. m ) and this time consider what happens when 4 6 (4 6) for 6 0. By the quotient rule our eponent for is -. But what does a negative eponent really mean? Let s do the same calculation long-hand by dividing the factors of 4 by the factors of So we see that to the power - is the same as one divided by to the power +. Here is the negative power rule for eponents. Negative Power Rule for Eponents n n 6 0 You will also see negative powers applied to products and fractions. For eample, here it is applied to a product. ( y) 6 y using the power rule 6 y 6 y 6 y using the negative power rule separately on each variable Here is an eample of a negative power applied to a quotient. a b a b a b a a b b a b a b a b a b using the power rule for quotients using the negative power rule on each variable separately simplifying the division of fractions using the power rule for quotients in reverse. The last step is not necessary but it helps define another rule that will save us time. A fraction to a negative power is flipped. n Negative Power Rule for Fractions y y n, 6 0,y 6 0 In some instances, it is more useful to write epressions without fractions and that makes use of negative powers. Eample Write the following epressions without fractions. (a) (b) (c) y (d) y 4

17 Chapter. Properties of Eponents We apply the negative rule for eponents n n on all the terms in the denominator of the fractions. (a) (b) (c) y y (d) y y Sometimes, it is more useful to write epressions without negative eponents. Eample Write the following epressions without negative eponents. (a) (b) a b c (c) 4 y (d) y We apply the negative rule for eponents n n on all the terms that have negative eponents. (a) (b) a b c a b c (c) 4 y 4y (d) y y Eample Simplify the following epressions and write them without fractions. (a) 4a b a 5 b (b) y y 4 (a) Reduce the numbers and apply quotient rule on each variable separately. (b) Apply the power rule for quotients first. 4a b 6a 5 b a 5 b a b y y 4 8 y 6 y 4 Then simplify the numbers, use product rule on the s and the quotient rule on the y s. 8 y 6 y 4 + y 6 5 y 5 5

18 .. Zero, Negative, and Fractional Eponents Eample 4 Simplify the following epressions and write the answers without negative powers. (a) ab b (b) y y (a) Apply the quotient rule inside the parenthesis. ab (ab 5 ) b Apply the power rule. (b) Apply the quotient rule on each variable separately. (ab 5 ) a b 0 a b 0 y y y ( ) 5 y 4 y4 5 Simplify Epressions With Fractional Eponents The eponent rules you learned in the last three sections apply to all powers. So far we have only looked at positive and negative integers. The rules work eactly the same if the powers are fractions or irrational numbers. Fractional eponents are used to epress the taking of roots and radicals of something (square roots, cube roots, etc.). Here is an emaple. p a a and p a a and 5p a a 5 a 5 a 5 Roots as Fractional Eponents mp a n a n m We will eamine roots and radicals in detail in a later chapter. In this section, we will eamine how eponent rules apply to fractional eponents. Eample 5 Simplify the following epressions. (a) a a (b) a (c) a 5 a (d) y (a) Apply the product rule. 6

19 Chapter. Properties of Eponents a a a + a 5 6 (b) Apply the power rule. a a (c) Apply the quotient rule. a 5 a a 5 a 4 a (d) Apply the power rule for quotients. y y Evaluate Eponential Epressions When evaluating epressions we must keep in mind the order of operations. You must remember PEMDAS. Evaluate inside the Parenthesis. Evaluate Eponents. Perform Multiplication and Division operations from left to right. Perform Addition and Subtraction operations from left to right. Eample 6 Evaluate the following epressions to a single number. (a) 5 0 (b) 7 (c) (d) (e) 6 (f) 8 (a) 5 0 Remember that a number raised to the power 0 is always. (b) (c) 8 7 (d) 7 (e) 6 p 6 4 Remember that an eponent of means taking the square root. 7

20 .. Zero, Negative, and Fractional Eponents (f) 8 8 Eample 7 p 8 Remember that an eponent of Evaluate the following epressions to a single number. (a) (b) 4 5 (c) 4 (a) Evaluate the eponent. means taking the cube root. Perform multiplications from left to right Perform additions and subtractions from left to right (b) Treat the epressions in the numerator and denominator of the fraction like they are in parenthesis. ( 4 5 ) ( ) ( 6 5) (9 4) ( 75) (c) Eample 8 Evaluate the following epressions for, y, z. (a) (b) ( y ) (c) y + 4z y 5 4z (a) y + 4z ( ) ( ) (b) ( y ) ( ( ) ) (4 ) 9 (c) y 5 4z ( ) ( ) ( ) Homework Problems Simplify the following epressions, be sure that there aren t any negative eponents in the answer. 8

21 Chapter. Properties of Eponents. y y 5 z 7 a 5. b 6. (a b c ) 7. Simplify the following epressions so that there aren t any fractions in the answer a (a 5 ) a y 8 y (4ab 6 ) (ab) 5 ( )(4 4 ) (y). a b c Evaluate the following epressions to a single number (6.) y 4 4y if and y. a 4 (b ) + ab if a and b. 5 y + z if, y, and z 4. a b if a 5 and b Review Answers y z 4. 7 y y 6. b a or b a 7. 7b 6 c 9 a a y. 7 y. y. 6 7 y 4. a b c 5. y

22 .. Zero, Negative, and Fractional Eponents