Simplifying Exponential Expressions

Transcription

1 Simplifying Eponential Epressions

2 Eponential Notation Base Eponent Base raised to an eponent Eample: What is the base and eponent of the following epression? 7 is the base 7 is the eponent

3 Goal To write simplified statements that contain distinct bases, one whole number in the numerator and one in the denominator, and no negative eponents. E: ab 9bc 6a b c 4a

4 Multiplying Terms When we are multiplying terms, it is easiest to break the problem down into steps. First multiply the number parts of all the terms together. Then multiply the variable parts together. Eamples: a. ( 4 )( -5 ) = ( )(. ) = -0 Only the z is squared b. (5z)(z)(4y) = (5... 4)(y. z. z) = 0yz

5 Eploration Evaluate the following without a calculator: 4 = = = = Describe a pattern and find the answer for: 0 = 8 7 9

6 Zero Power a 0 = Anything to the zero power is one Can a equal zero? No. You can t divide by 0.

7 Eploration Simplify: 4 Use the definition of eponents to epand 4 There are 7 variables Notice (from the initial epression) +4 is 7! 7

8 Product of a Power If you multiply powers having the same base, add the eponents. a m n

9 Simplify: Add the eponents since the bases are the same Eample 9 y 0 9 Anything raised to the 0 power is

10 Practice Simplify the following epressions: ) ) z y 5 y 4 ) y

11 Eploration Simplify: 5 The Product of a Power Rule says to add all the s Adding five times is equivalent to multiplying by 5. The same eponents from the initial epression! 5 5 Use the definition of eponents to epand

12 Power of a Power a mn To find a power of a power, multiply the eponents.

13 Eample 6 Simplify: Multiply the powers of a eponent raised to another power Any base without a power, is assumed to have an eponent of s s t 4t s s 6 t 9 ss t 4t 4 8s t 4t s t 9 Multiply numbers without eponents and add the eponents when the bases are the same

14 Practice Simplify the following epressions: ) y 4 8 y ) 5 a 4 a a ) y y 7 y

15 Eploration Simplify: z 5 z z z z z Adding five times is equivalent to multiplying by 5 Notice: Both the z and were raised to the 5 th power! z z The Product of a Power Rule says to add the eponents with the same bases Use the definition of eponents to epand

16 Power of a Product a m b m If a base has a product, raise each factor to the power

17 Eample Simplify: 4 5 Everything inside the parentheses is raised to the eponent outside the parentheses y y 5 y 0 0 y Multiply the powers of a eponent raised to another power y 45 Multiply numbers without eponents and add the eponents when the bases are the same

18 Practice Simplify the following epressions: ) pqr ab a pqr 4 5 ) 4 ) - yz 54 y z 9 8 5a b 7

19 First Four a 5 b d 6 0. r 8 s a 7 b 7 c. 6z 8. 9a m 6 n z y r 5 s 5 t 5 4. a 4 b 4 c 6. 08a 7. 7b y

20 Eploration Complete the tables (with fractions) by finding the pattern /5 /5 /5 / / /6 /8 ¼ ½ 4 8 6

21 Negative Powers A simplified epression has no negative eponents. a m a m Negative Eponents flip and become positive a m

22 Eample Simplify: 0 4 All of the old rules still apply for negative eponents Flip ONLY the thing with the negative eponent to the bottom and the eponent becomes positive 4a b 5a 45 a 0 4 0a b 0b a 6 b 6 This is not simplified since there is a negative eponent

23 Simplify: Everything with a positive eponent stays where it is. Eample 8y 8 y y y Everything with a negative eponent is flipped and eponent becomes positive. Since all of the negative eponents are gone, apply all of the old rules to simplify.

24 Practice Simplify the following epressions: ) 8 ) ) y y 8 4 4) a b 5 7 y y 7 8 a 4b 6

25 Eploration Simplify: 0 Use the definition of eponents to epand The 6 s in the denominator cancel 6 out of the 0 s in the numerator. This is the same as subtracting the eponents from the initial epression! Since everything is multiplied, you can cancel common factors Only 4 s remain in the numerator

26 Quotient of a Power a m n To find a quotient of a power, subtract the denominator s eponent from the numerator s eponent if the bases are the same. a 0

27 Eample Simplify: Divide the base numbers first 6 6 y 6y 6 y Subtract the eponents of the similar bases since there is division Not simplified since there is a negative eponents 4 y 4 y Flip any negative eponents

28 Practice Simplify the following epressions: 6 0 ab ) 5 a a 5 ) 4 y 6 y 5 ) 9 4y 4y 7 6 y

29 Eploration Simplify: Use the definition of eponents to epand Use the definition of eponents to rewrite. Notice: Both the numerator and denominator were raised to the 6 th power! a b a a a a a a b b b b b b aaaaaa bbbbbb 6 a 6 b 6 Multiply the fractions

30 Power of a Quotient a m b To find a power of a quotient, raise the denominator and numerator to the same power. m

31 Eample Simplify: Everything in the fraction is raised to the power out side the parentheses. Subtract the eponents when there is division, and add when there is multiplication 7 y y 5 y 6 8 y 5 y 7 y 5 y 8 y y y 9 9 Multiply the fractions

32 Practice Simplify the following epressions: ) ) ) a bc y s f zr 4 8 a b y f r s z