2Warm UpEvaluate each expression for the given values of the variables.1. x3y2 for x = –1 and y = 10for x = 4 and y = (–7)Write each number as a power of the given base.–1003. 64; base 4434. –27; base (–3)(–3)3
3Objectives Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents.
4You have seen positive exponents 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.PowerValue5554535251505–15–23125625125255 5 5 5 5
5When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.
8Notice the phrase “nonzero number” in the previous table 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.
92–4 is read “2 to the negative fourth power.” Reading Math
10Example 1: ApplicationOne cup is 2–4 gallons. Simplify this expression.cup is equal to
11Check It Out! Example 1A sand fly may have a wingspan up to 5–3 m. Simplify this expression.5-3 m is equal to
12Example 2: Zero and Negative Exponents Simplify.A. 4–3B. 70Any nonzero number raised to the zero power is 1.7º = 1C. (–5)–4D. –5–4
13In (–3)–4, the base is negative because the negative sign is inside the parentheses. In –3–4 the base (3) is positive.Caution
14Check It Out! Example 2Simplify.a. 10–4b. (–2)–4c. (–2)–5d. –2–5
15Example 3A: Evaluating Expressions with Zero and Negative Exponents Evaluate the expression for the given value of the variables.x–2 for x = 4Substitute 4 for x.Use the definition
16Example 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 = –3Substitute 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.
17Check It Out! Example 3aEvaluate the expression for the given value of the variable.p–3 for p = 4Substitute 4 for p.Evaluate exponent.Write the power in the denominator as a product.Evaluate the powers in the product.
18Check It Out! Example 3bEvaluate the expression for the given values of the variables.for a = –2 and b = 6Substitute –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.2Simplify.
19What if you have an expression with a negative exponent in a denominator, such as ? 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.
20Example 4: Simplifying Expressions with Zero and Negative Numbers A. 7w–4
21Example 4: Simplifying Expressions with Zero and Negative Numbers C.and
22Check It Out! Example 4Simplify.a. 2r0m–3rº = 1 andb.c.
23Lesson Quiz: Part I1. A square foot is 3–2 square yards. Simplify this expression.Simplify.2. 2–63. (–7)–34. 6015. –112–121
24Lesson Quiz: Part IIEvaluate each expression for the given value(s) of the variables(s).6.x–4 for x =107.for a = 6 and b = 3