Integral Exponents

Integral Exponents

1. Integral Exponents

2. Warm Up Evaluate each expression for the given values of the variables. 1. x 3 y 2 for x = –1 and y = 10 2. for x = 4 and y = (–7) Write each number as a power of the given base. –100 4 3 3. 64; base 4 ( – 3) 3 4. – 27; base ( – 3)

3. 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 625 125 25 5  5 Power Value 5 5 5 4 5 3 5 2 5 1 5 –1 5 5 –2  5  5  5

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

5. Base x Exponent Remember! 4

6. Notice the phrase “nonzero number” in the previous table. This is because 0 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. 2 –4 is read “2 to the negative fourth power.” Reading Math

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

9. Check It Out! Example 1 A sand fly may have a wingspan up to 5 –3 m. Simplify this expression. 5 -3 m is equal to

10. Example 2: Zero and Negative Exponents Simplify. A. 4 –3 B. 7 7º = 1 Any nonzero number raised to the zero power is 1. C. (–5) –4 D. –5 –4

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

12. Check It Out! Example 2 Simplify. a. 10 –4 b. (–2) –4 c. (–2) –5 d. –2 –5

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

14. Example 3B: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given values of the variables. –2 a 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.

15. Check It Out! Example 3a 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.

16. Check It Out! Example 3b 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.

17. 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.

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

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

20. Check It Out! Example 4 Simplify. a. 2 r m –3 b. c. rº = 1 and .

21. Lesson Quiz: Part I 1. A square foot is 3 –2 square yards. Simplify this expression. Simplify. 2. 2 –6 3 . (–7) –3 4. 6 5. –11 2 1 –121

22. Lesson Quiz: Part II Evaluate each expression for the given value(s) of the variables(s). 6. x –4 for x =10 7. for a = 6 and b = 3