Zero and negative exponent properties Lesson 5 exponents Zero and negative exponent properties Lesson 5

a m • a n = a m + n = a m – n a m a n Review What do we do when we want to multiply terms with the same base? What do we do when we want to divide terms with the same base? a m • a n = a m + n = a m – n a m a n

4 7 4 9 = 4 7−9 4 7 4 9 Practice The exponent is negative?!!! 4 7 4 9 4 7 4 9 = 4 7−9 =4 −2 Subtracted Exponents The exponent is negative?!!! Let’s come back to this problem in a few minutes...

Where do we Use negative exponents? The diameter of a human hair is about 10 −3 inches. 10 −3 is the same as 0.001 or 1 10 3 or 1 1,000

Where do we use negative exponents? Chameleons and geckos can climb smooth, vertical surfaces because of tiny hairs on their feet that are 10 −4 meters long. 10 −4 is the same as 0.0001 or 1 10,000 or 1 10 4

Negative and zero exponents So what would we do with the 4 −2 we got in our earlier problem? Let’s find out…

Negative and zero exponents So how should we correctly write 4 −2 using only positive exponents? 4 −2 = 1 4 2 = 1 16

Negative Exponent Property For any integer a, and any exponent n≠0, 𝑎 −𝑛 = 1 𝑎 𝑛 or 1 𝑎 −𝑛 = 𝑎 𝑛 Algebra Numbers 𝑎 −𝑛 = 1 𝑎 𝑛 or 1 𝑎 −𝑛 = 𝑎 𝑛 Words Negative Exponent Property 3 −8 = 1 3 8 or 1 3 −8 = 3 8

5 2 ∙ 5 −5 = 5 2+(−5) Product of Powers Property practice 5 2 ∙ 5 −5 5 2 ∙ 5 −5 = 5 2+(−5) Product of Powers Property = 5 −3 Adding Powers = 1 5 3 Definition of Negative Exponent = 1 125 Simplifying

4 𝑛 −11 4 𝑛 −11 =4∙ 𝑛 −11 Rewriting the Expression Practice 4 𝑛 −11 4 𝑛 −11 =4∙ 𝑛 −11 Rewriting the Expression = 4 1 ∙ 1 𝑛 11 Definition of Negative Exponents = 4 𝑛 11 Multiplying

1 49 1 16 1 −16 100 7 −2 −2 −4 − 2 −4 10 −5 ∙ 10 7 Come to the board! 7 −2 −2 −4 − 2 −4 10 −5 ∙ 10 7 1 3 2 4 1 49 1 16 1 −16 100

Come to the board! 5 𝑥 4 𝑥 7 10 2 𝑎 −4 𝑎 4 𝑏 2 ∙ 𝑏 −2 6∙ 6 −3 1 3 2 4 5 𝑥 3 100 𝑎 8 1 1 36