8.3 Zero, Negative, and Fractional Exponents

Transcription

1 Chapter 8. Eponents and Polynomials 8.3 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 m n m ) and consider what happens when nm. Lets take the eample of divided by. (4 4) 0 Now we arrived at the quotient rule by considering how the factors of cancel in such a fraction. Lets do that again with our eample of divided by. So 0. This works for any value of the eponent, not just 4. 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, 0 395

2 8.3. Zero, Negative, and Fractional Eponents Simplify Epressions With Negative Eponents Again we will look at the quotient rule for eponents (that n m n m ) and this time consider what happens when m>n. Lets take the eample of divided by 6. 6 (4 6) 2 for 0. By the quotient rule our eponent for is 2. But what does a negative eponent really mean? Lets do the same calculation long-hand by dividing the factors of by the factors of So we see that to the power 2 is the same as one divided by to the power +2. Here is the negative power rule for eponents. Negative Power Rule for Eponents n n 0 You will also see negative powers applied to products and fractions. For eample, here it is applied to a product. ( 3 y) 2 6 y 2 using the power rule 6 y 2 6 y 2 6 y 2 using the negative power rule separately on each variable Here is an eample of a negative power applied to a quotient. ( a b) 3 a 3 b 3 a 3 a 3 b 3 a 3 b3 b3 a 3 b 3 a 3 ( b a b 3 a 3 b3 using the power rule for quotients using the negative power rule on each variable separately simplifying the division of fractions ) 3 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. Negative Power Rule for Fractions( y) n ( y ) n, 0,y 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) 2 2 (c) 2 y 3 (d) 3 y 396

3 Chapter 8. Eponents and Polynomials We apply the negative rule for eponents n n on all the terms in the denominator of the fractions. (a) (b) (c) 2 y 3 2 y 3 (d) 3 y 3 y Sometimes, it is more useful to write epressions without negative eponents. Eample 2 Write the following epressions without negative eponents. (a) 3 3 (b) a 2 b 3 c (c) 4 y 3 (d) 2 2 y 3 We apply the negative rule for eponents n n on all the terms that have negative eponents. (a) (b) a 2 b 3 c a2 b 3 c (c) 4 y 3 4y3 (d) 2 2 y 3 2y3 2 Eample 3 Simplify the following epressions and write them without fractions. (a) 4a2 b 3 2a 5 b ) (b)( 3y 3 2 y 2 4 (a) Reduce the numbers and apply quotient rule on each variable separately. (b) Apply the power rule for quotients first. 4a 2 b 3 6a 5 b 2 a2 5 b 3 2a 3 b y y y 6 2 y 4 Then simplify the numbers, use product rule on the s and the quotient rule on the y s. 8 3 y 6 2 y y y 5 397

4 8.3. Zero, Negative, and Fractional Eponents Eample 4 Simplify the following epressions and write the answers without negative powers. (a) ab 2 2 b 3 (b) 3 y 2 2 y 2 (a) Apply the quotient rule inside the parenthesis. Apply the power rule. ab 2 2 (ab 5 ) 2 b 3 (b) Apply the quotient rule on each variable separately. (ab 5 ) 2 a 2 b 0 a2 b 0 3 y 2 2 y y 2 ( 2) 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. aa /2 and 3 aa /3 and 5 a 2 ( a 2) 5 a 2 5 a 2/5 Roots as Fractional Eponents m 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. 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)

5 Chapter 8. Eponents and Polynomials (c) (d) 3 3 (a) 5 0 Remember that a number raised to the power 0 is always. (b) (c) (d) Eample 7 Evaluate the following epressions to a single number. (a) (b) (c) ( ) (a) Evaluate the eponent. 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. ( ) ( ) ( ) (9 4) (32 75) (c) ( ) ( ) Eample 8 Evaluate the following epressions for 2,y, z 3. (a) 2 2 3y 3 + 4z (b)( 2 y 2 ) 2 (c) 3 2 y 5 2 4z 399

6 8.3. Zero, Negative, and Fractional Eponents (a) 2 2 3y + 4z ( ) ( ) (b) ( 2 y 2 ) 2 (2 2 ( ) 2 ) 2 (4 ) (c) 3 2 y 5 2 4z 3 22 ( ) 5 2 ( ( ) 2 ) 2 ( 2 2 ) 2 ( ) 2 ( ) 2 ( ) 2 Review Questions Simplify the following epressions, be sure that there aren t any negative eponents in the answer.. y y 5 z 7 5. ( 2 y 2 3)( 2 y 3) ( 6. a 2 b) 7. (3a 2 b 2 c 3 ) Simplify the following epressions so that there aren t any fractions in the answer a 3 (a 5 ) a y 2 8 y (4ab 6 ) 3 (ab) ( 5 ) 3 3 y /3 3 2 y 3/2 y /2 (3 3 )(4 ) (2y) 2 5. a 2 b 3 6. c /2 y 5/2 3/2 y 3/2 Evaluate the following epressions to a single number (6.2) (6 2) y 4 4y 2 if 2 and y 22. a 4 (b 2 ) 3 + 2ab if a 2 and b y 3 + 3z if 3, y2, and z4 24. a 2 2 b if a5 and b3 3 Review Answers 400 y

7 Chapter 8. Eponents and Polynomials z y /2 y ( /3 6. b 2 a) or b 2 a 2 27b 7. 6 c 9 a a y. 64a 2 b y 3. 3y y 2 5. a 2 b 3 c 6. y