Section 4.6 Negative Exponents

Transcription

1 Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals of fractions are relatively easy when the fraction is something like 3 5 : The reciprocal of 3 5 is 5 3. In other words, we simply invert the fraction, turn it over. However, the reciprocal of a whole number, such as 7, is the fraction 7. It s harder to see the inverted fraction because with 7 there doesn t seem to be any fraction at all. Of course, we need to think of 7 as 7 ; then it s easier to see its reciprocal: 7. Example : Find the reciprocal of each number or variable. a) 4 9 b) c) 6 d) - 8 e) x Answer: If it isn t a fraction already, think of it as being over, then invert the fraction. a) The reciprocal of 4 9 is 9 4 b) The reciprocal of is c) reciprocal of 6 is 6. d) reciprocal of - 8 is - 8. e) reciprocal of x is x. Exercise Find the reciprocal of each number or variable. a) Reciprocal of 2 9 is b) Reciprocal of is c) Reciprocal of 4 is d) Reciprocal of - 5 is e) Reciprocal of w is f) Reciprocal of - y is Negative Exponents page 4.6 -

2 UNDERSTANDING NEGATIVE EXPONENTS Remember this: Exponents have more meaning than they have value. In this section we re going to look at negative exponents, such as x -2, y -5, and 2-3. However, when the exponent is negative, such as 2-3, we can t really say that there are negative three factors of 2. Instead, we need to find out the meaning of the negative in an exponent, because exponents have more meaning than they have value. There are several ways that we can develop the idea of a negative exponent, a few of which will be shown here. Showing you a variety of methods is not intended to confuse you; instead it is intended to cement the meaning of the negative exponent, to give it greater validity. Of course, we will be developing a rule, the Rule of Negative Exponents. First, let s start with something rather familiar to us: powers of 3. We know that 3 0 and that 3 3 and that multiplying by another increases the exponent by multiplying by another increases the exponent by multiplying by another increases the exponent by multiplying by another increases the exponent by and so on. We ll now consider working in the reverse direction, starting with and divide by 3, thereby decreasing the exponent by. (Dividing by 3 will mean one less factor of 3.) Of course, dividing by 3 is the same as multiplying by 3. So when the process takes us down to, we can think of multiplying by 3 : 3 3. Furthermore, we can multiply 3 by 3 and get 9 ; we can then multiply 9 by 3 and get 27. and on and on. Now consider going from one power of 3 to the next by dividing by 3 and let s take a close look at the progression of the exponent: Negative Exponents page 4.6-2

3 Start with divide by 3 ( or multiply by 3 ) 8 34 and decrease the exponent by divide by 3 ( or multiply by 3 ) and decrease the exponent by divide by 3 ( or multiply by 3 ) and decrease the exponent by divide by 3 ( or multiply by 3 ) 3 3 and decrease the exponent by divide by 3 ( or multiply by 3 ) 3 0 and decrease the exponent by divide by 3 ( or multiply by 3 ) divide by 3 ( or multiply by 3 ) divide by 3 ( or multiply by 3 ) and decrease the exponent by and decrease the exponent by and decrease the exponent by Exercise 2 divide by 3 ( or multiply by 3 ) and so on and decrease the exponent by Duplicate the process above using a base of 2; follow the outline given. Start with divide by 2 ( or multiply by 2 ) 6 24 and decrease the exponent by divide by 2 ( or multiply by 2 ) divide by 2 ( or multiply by 2 ) divide by 2 ( or multiply by 2 ) divide by 2 ( or multiply by 2 ) divide by 2 ( or multiply by 2 ) divide by 2 ( or multiply by 2 ) divide by 2 ( or multiply by 2 ) and decrease the exponent by and decrease the exponent by and decrease the exponent by and decrease the exponent by and decrease the exponent by and decrease the exponent by and decrease the exponent by Negative Exponents page 4.6-3

4 Let s look graphically at the powers of 2 you just generated. This graph will not be to scale. In fact, the identity for multiplication,, will be treated as the center. Notice that all of the powers of 2 are to the right of 0. That means that they are all positive. Also notice that all of the values between 0 and, such as 6, 8, 4 and 2, are reciprocals. Please note: 2-3 does not mean - 8, nor does it mean - 8 exponent has its own meaning, reciprocal.. The negative in the THE RULE OF NEGATIVE EXPONENTS Here is the Rule of Negative Exponents; The Rule of Negative Exponents: For any non-zero real number x, x - n x n. in other words, x neg. n means the reciprocal of x n. Example 2: Rewrite each expression with a positive exponent. Evaluate if possible. a) 4-3 b) 5-2 c) 7 - d) x - Answer: Apply the Rule of Negative Exponents. The meaning of the negative in an exponent is reciprocal. a) 4-3 c) b) d) x x x Negative Exponents page 4.6-4

5 Exercise 3 Rewrite each expression with a positive exponent. Evaluate if possible. a) 2-4 b) 3-3 c) 7-2 d) 9-2 e) 6 - f) 0-3 g) x - 4 h) y - 5 i) w - j) a - 9 k) m - l) c - 2 THE NEGATIVE EXPONENT AND FRACTIONS At the beginning of this section we were reminded about reciprocals; that is because, as we now know, the negative in the exponent means reciprocal. This is even more evident when the base is a fraction, such as If we first think of this as 3 neg. two, then we can use the meaning of the negative in the exponent to rewrite this as its reciprocal raised to a positive power: distributive rules of exponents: two 2. We can then evaluate this further using one of the Notice in going from 2 neg. two 3 to 3 two 2, two things happened: () we eliminated the negative in the exponent and (2) we inverted the fraction Actually, what happened was, applying the negative exponent to the fraction caused the fraction to invert; that s because the negative part of the exponent means reciprocal. Negative Exponents page 4.6-5

6 Example 3: Rewrite each expression with a positive exponent. Evaluate if possible. a) b) c) 6-5 d) x Answer: Apply the Rule of Negative Exponents. The meaning of the negative in an exponent is reciprocal. Also apply the one of the rules of distribution. a) b) c) d) x x 25 x 5 32 x 5 Exercise 4 Rewrite each expression with a positive exponent. Evaluate if possible. a) b) c) 8 - d) e) 6 - v f) x g) 9-2 5m h) 2x - 4 w Negative Exponents page 4.6-6

7 ANOTHER WAY TO DEVELOP NEGATIVE EXPONENTS In the Section 4. we saw the quotient rule for exponents: x a x b x a b. This rule is easy to apply for x 5 something like x 3 x 5 3 x 2. It s also easy to understand when we expand the numerator and denominator and cancel common factors: The numerator had five factors of x, but we canceled three of them, leaving it with only two factors of x. What would happen, though, if the denominator had more factors than the numerator? In other words, what would happen to something like x3 x 5? If we were to expand it and cancel, as before, we would get: Yet, if we were, instead, to use the quotient rule, we d get x 3 x 5 x 3 5 x - 2. So, in one way x 3 x 5 x 2 and in another way x 3 x 5 x - 2. Therefore, x - 2 x 2. Using words and meaning, this says x neg. two x two ; the negative means reciprocal and the exponent two means two factors of x. This time, though, the two factors of x are in the denominator. USING NEGATIVE EXPONENTS WITH THE OTHER RULES. We can apply the rules learned in Section 4. to negative exponents as well as positive exponents. As a standard, though, it is often request that the end result be written with positive exponents only. Consider these examples: the product rule: x 5 x - 2 x 5 + (- 2) x 3. the quotient rule: x 5 x - 2 x 5 (- 2) x x 7. the power rule: ( x 3 ) - 4 x 3(- 4) x - 2 x 2 Negative Exponents page 4.6-7

8 With negative numbers of any kind, we need to be careful, and that is especially true when working with the rules of exponents. For the sake of accuracy, it is recommended that you do all of the steps and show all of your work. Also, think, think, think! Example 4: Simplify each expression. Be sure to write the result with positive exponents only. a) x 7 x - 3 b) w - 2 w - 5 c) c 3 c - 4 d) x - 3 x - e) ( a - 2 ) - 3 f) ( p - 5 ) 2 Answer: Carefully apply the rules of exponents. a) x 7 x - 3 x 7 + (- 3) x 4 b) w - 2 w - 5 w (- 5) w - 7 w 7 c) c 3 c - 4 c 3 (- 4) c c 7 d) x - 3 x - x - 3 (- ) x x - 2 x 2 e) ( a - 2 ) - 3 a - 2(- 3) a 6 f) ( p - 5 ) 2 p - 5(2) p - 0 p 0 Exercise 5 Simplify each expression. Be sure to write the result with positive exponents only. a) x 9 x - 2 b) v - 4 v 5 c) y 2 y - 8 d) w - 3 w - 6 e) a a - 6 f) m - 4 m - 7 g) x - 6 x - 2 h) y - y 5 i) ( a - 7 ) - 2 j) ( w - ) - 3 k) ( p - 2 ) 6 l) ( m 4 ) - Negative Exponents page 4.6-8

9 Answers to each Exercise Section 4.6 Exercise : a) e) 9 2 b) c) 4 d) - 5 w f) - y Exercise 2: i) Exercise 3: ii) a) iii) 2 2 c) iv) 2 0 e) v) vi) vii) g) i) k) b) d) f) 0 3,000 x 4 h) y 5 w w j) a 9 m m l) c 2 Exercise 4: a) b) ,000 c) 8 8 d) e) v 6 v 6 f) 4 2 x 6 x 2 g) 5m m2 8 h) w 4 2x w 4 6x 4 Exercise 5: a) x 7 b) v or v c) y - 6 y 6 d) w - 9 e) a 7 f) m 3 g) x - 4 x 4 h) y - 6 y 6 i) a 4 j) w 3 k) p - 2 p 2 l) m - 4 w 9 m 4 Negative Exponents page 4.6-9

10 Section 4.6 Focus Exercises. Rewrite each expression with a positive exponent. (Remember the meaning of the negative exponent.) Evaluate if possible. a) 8 - b) 3-4 c) - 2 d) x - e) y - 2 f) w Rewrite each expression with positive exponents only. (Remember the meaning of the negative exponent.) Evaluate if possible. a) 3-5 b) c) 2-2 d) b - 5 c e) - 3 4m f) 2x - 4 w 3. Simplify each expression. Be sure to write the result with positive exponents only. a) m 4 m - 3 b) p - 7 p 6 c) k 9 k - 3 d) h - 8 h - 5 e) y y - 5 f) m - 7 m - 4 g) m - 5 m - 9 h) k - k 6 i) ( x - ) 5 j) ( m 7 ) - k) ( h - ) - 8 l) ( y - 4 ) - 9 Negative Exponents page 4.6-0