5A power is an expression written with an exponent and a base or the value of such an expression. 3² is an example of a power.The base is thenumber that isused as a factor.32The exponent, 2 tellshow many times thebase, 3, is used as afactor.
6When a number is raised to the second power, we usually say it is “squared.” The area of a square is s s = s2, is the side length.SWhen a number is raised to the third power, we usually say it is “cubed.” The of volume of a cube is s s s = s3 is the side length.S
7Example 1A: Writing Powers for Geometric Models Write the power represented by the geometric model.The figure is 5 units long, 5 units wide, and 5 units tall. 5 5 5553The factor 5 is used 3 times.
8Example 1B: Writing Powers for Geometric Models Write the power represented by the geometric model.6The figure is 6 units long and 6 units wide. 6 x 6662The factor 6 is used 2 times.
9Check It Out! Example 1Write the power represented by each geometric model.a.The figure is 2 units long and 2 units wide. 2 222The factor 2 is used 2 times.xb.The figure is x units long, x units wide, and x units tall. x x xxThe factor x is used 3 times.x3
10There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or a base and exponent.Reading ExponentsWordsMultiplicationPowerValue3 to the first power33133 to the second power, or 3 squared3 33293 to the third power, or 3 cubed3 3 333273 to the fourth power343 3 3 3813 to the fifth power3 3 3 3 335243
11Caution!In the expression –52, 5 is the base because the negative sign is not in parentheses. In the expression (–2), –2 is the base because of the parentheses.
12Example 2: Evaluating Powers Evaluate each expression.A. (–6)3(–6)(–6)(–6)Use –6 as a factor 3 times.–216B. –102Think of a negative sign in front of a power as multiplying by a –1.–1 • 10 • 10Find the product of –1 andtwo 10’s.–100
13Example 2: Evaluating Powers Evaluate the expression.C.29Use as a factor 2 times.29=48129
14Check It Out! Example 2Evaluate each expression.a. (–5)3(–5)(–5)(–5)Use –5 as a factor 3 times.–125b. –62Think of a negative sign in front of a power as multiplying by –1.–1 6 6Find the product of –1 andtwo 6’s.–36
15Check It Out! Example 2Evaluate the expression.c.Use as a factor 3 times.342764
16Example 3: Writing Powers Write each number as a power of the given base.A. 64; base 88 8The product of two 8’s is 64.82B. 81; base –3(–3)(–3)(–3)(–3)The product of four –3’s is 81.(–3)4
17Check It Out! Example 3Write each number as a power of a given base.a. 64; base 44 4 4The product of three 4’s is 64.43b. –27; base –3(–3)(–3)(–3)The product of three (–3)’s is –27.–33
18Example 4: Problem-Solving Application In case of a school closing, the PTApresident calls 3 families. Each ofthese families calls 3 other familiesand so on. How many families will havebeen called in the 4th round of calls?Understand the problem1The answer will be the number of familiescontacted in the 4th round of calls.List the important information:• The PTA president calls 3 families.• Each family then calls 3 more families.
19Example 4 Continued2Make a PlanDraw a diagram to show the number ofFamilies called in each round of calls.PTA President1st round of calls2nd round of calls
20Notice that after each round of calls the Example 4 ContinuedSolve3Notice that after each round of calls thenumber of families contacted is a power of 3.1st round of calls: 1 3 = 3 or 31 families contacted2nd round of calls: 3 3 = 9 or 32 families contacted3rd round of calls: 9 3 = 27 or 33 families contactedSo, in the 4th round of calls, 34 families will havebeen contacted.34 = 3 3 3 3 = 81Multiply four 3’s.In the fourth round of calls, 81 familieswill have been contacted.
21Example 4 ContinuedLook Back4Drawing a diagram helps you visualize theproblem, but the numbers become toolarge for a diagram. The diagram helps yourecognize the pattern of multiplying by 3so that you can write the number as apower of 3.
22Check it Out! Example 4What if…? How many bacteria will be on the slide after 8 hours?After each hour, the number of bacteria is a power of 2.282 2 2 2 2 2 2 2Multiply eight 2’s.256The product of eight 2’s.
23n n Lesson Quiz 1. Write the power represented by the geometric model. Simplify each expression.3. –63−2162.4. 62165. (–2)664Write each number as a power of the given base.; base 7737. 10,000; base 10104