Understanding Zero and Negative Exponents in Math

1. Objectives Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents.

2.  5  5  5  5 You have seen positive exponents. Recall that to simplify 32, use 3 as a factor 2 times: 32 = 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. 55 54 53 52 51 50 5–1 5–2 625 3125 125 25 5

3. When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.

4. Remember! Base x Exponent 4

6. Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = . Also 0–6 would be = . Since division by 0 is undefined, neither value exists.

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

8. Example 1: Application One cup is 2–4 gallons. Simplify this expression.

9. Your Turn 1 A sand fly may have a wingspan up to 5–3 m. Simplify this expression.

10. Example 2: Zero and Negative Exponents Simplify. A. 4–3 B. 70 C. (–5)–4 D. –5–4

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

12. Example 3A: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given value of the variables. x–2 for x = 4

13. Example 3B: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given values of the variables. –2a0b-4 for a = 5 and b = –3

14. Your Turn 3a Evaluate the expression for the given value of the variable. p–3 for p = 4

15. for a = –2 and b = 6 Your Turn 3b Evaluate the expression for the given values of the variables. 2

16. 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. An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents. 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.

17. B. Example 4: Simplifying Expressions with Zero and Negative Numbers Simplify. A. 7w–4

18. Example 4: Simplifying Expressions with Zero and Negative Numbers Simplify. C.

19. Your Turn 4 Simplify. a. 2r0m–3 b. c.

20. Objectives Evaluate and multiply by powers of 10. Convert between standard notation and scientific notation.

21. Vocabulary scientific notation

22. The table shows relationships between several powers of 10. Each time you divide by 10, the exponent decreases by 1 and the decimal point moves one place to the left.

23. The table shows relationships between several powers of 10. Each time you multiply by 10, the exponent increases by 1 and the decimal point moves one place to the right.

25. Example 1: Evaluating Powers of 10 Find the value of each power of 10. C. 109 A. 10–6 B. 104

26. Writing Math You may need to add zeros to the right or left of a number in order to move the decimal point in that direction.

27. Your Turn 1 Find the value of each power of 10. b. 105 c. 1010 a. 10–2 0.10 1,000,000 100,000,000,000

28. Reading Math If you do not see a decimal point in a number, it is understood to be at the end of the number.

29. Example 2: Writing Powers of 10 Write each number as a power of 10. B. 0.0001 C. 1,000 A. 1,000,000

30. Your Turn 2 Write each number as a power of 10. b. 0.0001 c. 0.1 a. 100,000,000 10-2 107 10-5

31. Multiplying by Powers of 10 You can also move the decimal point to find the value of any number multiplied by a power of 10. You start with the number rather than starting with 1.

32. Example 3: Multiplying by Powers of 10 Find the value of each expression. A. 23.89  108 B. 467  10–3

33. Your Turn 3 Find the value of each expression. a. 853.4  105 85,340,000 b. 0.163  10–2 0.00163

34. Scientific notation is a method of writing numbers that are very large or very small. A number written in scientific notation has two parts that are multiplied. The first part is a number that is greater than or equal to 1 and less than 10. The second part is a power of 10.

35. Example 4A: Astronomy Application Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km. Write Saturn’s diameter in standard form. Write Saturn’s distance from the Sun in scientific notation.

36. Reading Math Standard form refers to the usual way that numbers are written—not in scientific notation.

37. Example 5: Comparing and Ordering Numbers in Scientific Notation Order the list of numbers from least to greatest. Step 1 List the numbers in order by powers of 10. Step 2 Order the numbers that have the same power of 10

38. Your Turn Example 5 Order the list of numbers from least to greatest. Step 1 List the numbers in order by powers of 10. Step 2 Order the numbers that have the same power of 10

39. Homework Pg. 449: 24 – 57 odd Pg. 455: 14 – 35 odd