1. Zero and Negative Exponents Section 7-1

2. Goals Goal To simplify expressions involving zero and negative exponents. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary None

4. Reciprocals When working with negative exponents you need to know what a reciprocal is… We already covered this earlier in the course so as a quick review… A reciprocal is a fraction that is inverted and the product is 1. It looks like this: 2 6 2 6 OriginalReciprocal Example: Product ● =1

5. Power Value You have seen positive exponents. Recall that to simplify 3 2, use 3 as a factor 2 times: 3 2 = 3  3 = 9. But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out. 3125 625 125255  5 5 5 5454 5353 5252 5151 5 –1 5050 5 –2  5 5  5 5  5 5 Zero and Negative Exponents 11/5 1/25 When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.

6. Base x Exponent Remember! 4

7. DEFINITION OF ZERO EXPONENTS a 0 = 1, A nonzero number to the zero power has the value of ONE. ANY NUMBER (except zero!) RAISED TO THE ZERO POWER IS ONE!!!! a 0 = 1, where a is not equal to 0. 5 0 = 1 97 0 = 1 1,000,000 0 = 1 1,567 0 = 1 765 0 = 1 8 0 = 1 100 0 = 1 -22 0 = 1 -4 0 = 1 -9 0 = 1 1987 0 = 1 22.879 0 = 1 300 0 = 1 Definition

8. Zero Exponents Why is 0 0 undefined? The property of zero as an exponent implies the following pattern. 3 0 = 1 2 0 = 1 1 0 = 1 0 0 = 1 However, switching the exponent and the base implies the following pattern. 0 3 = 0 0 2 = 0 0 1 = 0 0 0 = 0 It is not possible for 0 0 to equal both 1 and 0. So, 0 0 is undefined. ? ?

9. What this really means is that you turn a negative exponent into a positive exponent by shifting the power from numerator to denominator or vice versa! a -n is the reciprocal of a n Definition DEFINITION OF NEGATIVE EXPONENTS, where a is not equal to 0.

10. 2 –4 is read “2 to the negative fourth power.” Reading Math

11. What this really means is that you turn a negative exponent into a positive exponent by shifting the power from numerator to denominator or vice versa! Important – only move the base number and the negative exponent that goes with it! Negative Exponents

12. Why is an expression with a base of 0 and a negative exponent undefined? Using 0 as a base with a negative exponent will result in division by zero, which is undefined.

13. Negative Exponents What does 2 -1 Mean? You cannot leave an exponent negative because there is no way to express it’s meaning. You must make it positive ! An algebraic expression is in simplest form when powers with a variable base are written with only positive exponents.

14. One cup is 2 –4 gallons. Simplify this expression. cup is equal to Example: Application

15. A sand fly may have a wingspan up to 5 –3 m. Simplify this expression. 5 -3 m is equal to Your Turn:

16. Simplify. A. 4 –3 B. 7 0 7º = 1 Any nonzero number raised to the zero power is 1. C. (–5) –4 D. –5 –4 Example: Zero & Negative Exponents

17. In (–3) –4, the base is negative because the negative sign is inside the parentheses. In –3 –4 the base (3) is positive. Caution

18. Simplify. a. 10 –4 b. (–2) –4 c. (–2) –5 d. –2 –5 Your Turn:

19. Evaluate the expression for the given value of the variables. x –2 for x = 4 Substitute 4 for x. Use the definition Example:

20. Evaluate the expression for the given values of the variables. –2a 0 b -4 for a = 5 and b = –3 Substitute 5 for a and –3 for b. Evaluate expressions with exponents. Write the power in the denominator as a product. Evaluate the powers in the product. Simplify. Example:

21. Evaluate the expression for the given value of the variable. p –3 for p = 4 Substitute 4 for p. Evaluate exponent. Write the power in the denominator as a product. Evaluate the powers in the product. Your Turn:

22. Evaluate the expression for the given values of the variables. for a = –2 and b = 6 2 Substitute –2 for a and 6 for b. Evaluate expressions with exponents. Write the power in the denominator as a product. Evaluate the powers in the product. Simplify. Your Turn:

23. What if you have an expression with a negative exponent in a denominator, such as ? or Definition of a negative exponent. Substitute –8 for n. Simplify the exponent on the right side. So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator. An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents. More Negative Exponents

24. Simplify. A. 7w –4 B. Example:

25. Simplify. C. and Example:

26. Simplify. a. 2r 0 m –3 b. c. rº = 1 and. Your Turn:

27. Review

28. Joke Time What washes up on tiny beaches? Microwaves! What’s the best way to crave wood? Whittle by whittle! What do you do when you see a spaceman? Park your car, man.

29. Assignment 7-1 Exercises Pg. 444 - 446: #10 – 54 even, 60 – 66 even.