Integral Exponents
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This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
Integral Exponents
- 2. Warm Up Evaluate each expression for the given values of the variables. 1. x3y2 for x = –1 and y = 10 –100 2. for x = 4 and y = (–7) Write each number as a power of the given base. 3. 64; base 4 43 4. –27; base (–3) (–3)3
- 3. 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. Power 55 54 53 52 51 50 5–1 5–2 3125 625 125 25 5 Value 5 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. Remember! Base x4 Exponent
- 7. 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.
- 8. Reading Math 2–4 is read “2 to the negative fourth power.”
- 9. Example 1: Application One cup is 2–4 gallons. Simplify this expression. cup is equal to
- 10. 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
- 11. Example 2: Zero and Negative Exponents Simplify. A. 4–3 B. 70 Any nonzero number raised to the zero power is 1. 7º = 1 C. (–5)–4 D. –5–4
- 12. Caution In (–3)–4, the base is negative because the negative sign is inside the parentheses. In –3–4 the base (3) is positive.
- 13. Check It Out! Example 2 Simplify. a. 10–4 b. (–2)–4 c. (–2)–5 d. –2–5
- 14. 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
- 15. 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 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.
- 16. 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.
- 17. Check It Out! Example 3b Evaluate the expression for the given values of the variables. for a = –2 and b = 6 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. 2
- 18. 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 ifexpression that contains negativeis in a denominator, it is equivalent to the same An a base with a negative exponent or zero exponents is not considered to be base with the opposite (positive) exponent in the numerator. exponents. simplified. Expressions should be rewritten with only positive
- 19. Example 4: Simplifying Expressions with Zero and Negative Numbers Simplify. A. 7w–4 B.
- 20. Example 4: Simplifying Expressions with Zero and Negative Numbers Simplify. C. and
- 21. Check It Out! Example 4 Simplify. a. 2r0m–3 rº = 1 and . b. c.
- 22. Lesson Quiz: Part I 1. A square foot is 3–2 square yards. Simplify this expression. Simplify. 2. 2–6 3. (–7)–3 4. 60 1 5. –112 –121
- 23. 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