1. 7-1 Integer Exponents Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 7-1 Integer Exponents 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 4343 3. 64; base 4 (–3) 3 4. –27; base (–3)

3. 7-1 Integer Exponents 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. California Standards

4. 7-1 Integer Exponents 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 make a conjecture. 3125 625 125255  5 5 Power Value 5 5454 5353 5252 5151 5 –1 5050 5 –2  5 5  5 5 5 5

5. 7-1 Integer Exponents When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.

6. 7-1 Integer Exponents x Remember! 4 Exponent Base

7. 7-1 Integer Exponents

8. 7-1 Integer Exponents Notice the phrase “nonzero number” in the previous table. This is because 0 0 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above in the table with a base of 0 instead of 5, you would get 0 0 =. Also, 0 –6 would be =. Since division by 0 is undefined, neither value exists.

9. 7-1 Integer Exponents 2 –4 is read “2 to the negative fourth power.” Reading Math

10. 7-1 Integer Exponents Additional Example 1: Measurement Application One cup is 2 –4 gallons. Simplify this expression. gal is equal to

11. 7-1 Integer Exponents 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

12. 7-1 Integer Exponents Additional Example 2: Zero and Negative Exponents 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

13. 7-1 Integer Exponents In (–3) –4, the base is negative because the negative sign is inside the parentheses. In –3 –4 the base (3) is positive. Caution

14. 7-1 Integer Exponents Check It Out! Example 2 Simplify. a. 10 –4 b. (–2) –4 c. (–2) –5 d. –2 –5

15. 7-1 Integer Exponents Additional Example 3A: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given value of the variable. x –2 for x = 4 Substitute 4 for x. Use the definition

16. 7-1 Integer Exponents Additional Example 3B: Evaluating Expressions with Zero and Negative Exponents 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.

17. 7-1 Integer Exponents 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.

18. 7-1 Integer Exponents 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.

19. 7-1 Integer Exponents 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. 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.

20. 7-1 Integer Exponents An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents. 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.

21. 7-1 Integer Exponents Simplify. Additional Example 4: Simplifying Expressions with Zero and Negative Numbers A. 7w –4 B.

22. 7-1 Integer Exponents Simplify. Additional Example 4: Simplifying Expressions with Zero and Negative Numbers C. and

23. 7-1 Integer Exponents Check It Out! Example 4 Simplify. a. 2r 0 m –3 b. c. r 0 = 1 and.

24. 7-1 Integer Exponents 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 0 5. –11 2 1 –121

25. 7-1 Integer Exponents 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 Simplify. 8. 4y –5 10. 9. –3y 6