Properties of Expoets
Defiitio Defiitio: Expoet Defiitio of a Expoet For ay coutig umber, b = bbb b N factors of b We refer to b at the power; the th power of b, or b raised to the th power. We call b the base ad the expoet. Slide 2
Defiitio Two powers of b have specific ames. We refer to b 2 as the square of b or b squared. We refer to b 3 as the cube of b or b cubed. Clarify Defiitio: Expoet Defiitio of a Expoet For b, we compute b before fidig the opposite. For 2 4, the base is 2, ot 2. If we wat the base 2 Slide 3
Calculator Defiitio: Expoet Defiitio of a Expoet Use a graphig calculator to check both computatios To fid 2 4, press ( ) 2 ^ 3 ENTER Slide 4
Properties Properties of Expoets Properties of Expoet Slide 5
Example Show that b 5 b 3 = b 5. Solutio Properties of Expoets Properties of Expoet Writig b 5 b 3 without expoets, we see that Use calculator to verify by usig various bases ad examiig the table Slide 6
Properties of Expoets Solutio Cotiued Properties of Expoet Example Show that b m b = b m+, where m ad are coutig umbers. Slide 7
Solutio Properties of Expoets Properties of Expoet Write b m b without expoets: Example Show that c 0. b = c b c, is a coutig umber ad Slide 8
Properties of Expoets Properties of Expoet Solutio Write b c without expoets: Slide 9
Simplifyig Expressios Ivolvig Expoets Property Simplifyig Expressios Ivolvig Expoets A expressio ivolvig expoets is simplified if 1. It icludes o paretheses. 2. Each variable or costat appears as a base as few times as possible. For example, we write x 2 x 4 = x 6 3. Each umerical expressio (such as 7 2 ) has bee calculated, ad each umerical fractio has bee simplified. 4. Each expoet is positive. Slide 10
Simplifyig Expressios Ivolvig Expoets Example Simplify. ( 2 3 bc) 1. 2 Simplifyig Expressios Ivolvig Expoets 5 ( 3 4)( 2 2 bc bc ) 2. 3 2 3bc 3. 12 bc 7 6 2 5 7 8 24bc 4. 16bcd 2 5 3 4 Slide 11
Simplifyig Expressios Ivolvig Expoets Solutio Simplifyig Expressios Ivolvig Expoets Slide 12
Simplifyig Expressios Ivolvig Expoets Simplifyig Expressios Ivolvig Expoets Solutio Cotiued Slide 13
Simplifyig Expressios Ivolvig Expoets Warig Simplifyig Expressios Ivolvig Expoets 3b 2 ad (3b) 2 are ot equivalet 3b 2 base is b, ad (3b) 2 base is the 3b Typical error looks like Slide 14
Simplifyig Expressios Ivolvig Expoets Zero as a Expoet Itroductio What is the meaig of b 0? The property is to be true for m =, the b b m = b m b 0 1 = = b = b, b 0 b So, a reasoable defiitio of b 0 is 1. Defiitio For b 0, b 0 = 1 Slide 15
Simplifyig Expressios Ivolvig Expoets Illustratio Zero as a Expoet 7 0 = 1, ( 3) 0 = 1, ad (ab) 0 = 1, where ab 0 Slide 16
Itroductio Negative Expoets Negative Expoets If is a egative iteger, what is the meaig of b? What is the meaig of a egative expoet? If the m property b m = b is true for m = 0, the b 0 1 b 0 = = b = b, b 0 b b So, we would defie b 1 to be. b Slide 17
Defiitio If b 0 ad is a coutig umber, the 1 b = b I words: To fid b, take its reciprocal ad switch the sig of the expoet. Illustratio For example Negative Iteger Expoets Negative Expoets 1 1 1 3 = = ad b = 3 9 b 2 5 2 5 Slide 18
Itroductio Negative Expoets Negative Expoets 1 We write i aother form, where b 0 ad is a b coutig umber: Slide 19
Defiitio Negative Expoets Negative Expoets If b 0 ad is a coutig umber, the 1 = b b I words: To fid 1, take its reciprocal ad switch t he sig of the expoet. b Example 4 8 For example, 1 1 = 28 = 16 ad = b. 4 8 2 b Slide 20
Simplifyig More Expressios Ivolvig Expoets Example Simplify More Expressios Ivolvig Expoets Simplify. 1. 9b 5 2. b 3. 3 4 Solutio 7 1 1 + 3 Slide 21
Properties Properties of Iteger Expoets Simplify More Expressios Ivolvig Expoets Slide 22