Negative Exponent a n = 1 a n, where a 0. Power of a Power Property ( a m ) n = a mn. Rational Exponents =

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1 Refeece Popetie Popetie of Expoet Let a ad b be eal umbe ad let m ad be atioal umbe. Zeo Expoet a 0 = 1, wee a 0 Quotiet of Powe Popety a m a = am, wee a 0 Powe of a Quotiet Popety ( a b m, wee b 0 b) m = am Negative Expoet a = 1 a, wee a 0 Powe of a Powe Popety ( a m ) = a m Ratioal Expoet a m/ = ( a 1/ ) m = ( a ) m Poduct of Powe Popety a m a = a m + Powe of a Poduct Popety (ab) m = a m b m Ratioal Expoet a m/ 1 = a m/ = 1 ( a 1/ ) m = 1 ( a ) m, wee a 0 Popetie of Radical Let a ad b be eal umbe ad let be a itege geate ta 1. Poduct Popety of Radical ab = a b Squae Root of a Negative Numbe 1. If i a poitive eal umbe, te = i.. By te fit popety, it follow tat ( i ) =. Quotiet Popety of Radical a b = a, wee a 0 ad b 0 b Refeece Popetie of Logaitm Let b, m, ad be poitive eal umbe wit b 1. Poduct Popety log b m = log b m + log b Quotiet Popety log m b = log b m log b Powe Popety log b m = log b m Ote Popetie Zeo-Poduct Popety If A ad B ae expeio ad AB = 0, te A = 0 o B = 0. Popety of Equality fo Expoetial Equatio If b > 0 ad b 1, te b x = b y if ad oly if x = y. Popety of Equality fo Logaitmic Equatio If b, x, ad y ae poitive eal umbe wit b 1, te log b x = log b y if ad oly if x = y. Refeece A97

2 Patte Squae of a Biomial Patte (a + b) = a + ab + b (a b) = a ab + b Cube of a Biomial (a + b) 3 = a 3 + 3a b + 3ab + b 3 (a b) 3 = a 3 3a b + 3ab b 3 Diffeece of Two Squae Patte a b = (a + b)(a b) Sum of Two Cube a 3 + b 3 = (a + b) ( a ab + b ) Sum ad Diffeece Patte (a + b)(a b) = a b Completig te Squae x + bx + ( b ) = ( x + ) b Pefect Squae Tiomial Patte a + ab + b = (a + b) a ab + b = (a b) Diffeece of Two Cube a 3 b 3 = (a b) ( a + ab + b ) Teoem Te Remaide Teoem If a polyomial f (x) i divided by x k, te te emaide i = f (k). Te Facto Teoem A polyomial f (x) a a facto x k if ad oly if f (k) = 0. Te Ratioal Root Teoem If f (x) = a x a 1 x + a 0 a itege coefficiet, te evey atioal olutio of f (x) = 0 a te fom p q = facto of cotat tem a 0. facto of leadig coefficiet a Te Iatioal Cojugate Teoem Let f be a polyomial fuctio wit atioal coefficiet, ad let a ad b be atioal umbe uc tat b i iatioal. If a + b i a zeo of f, te a b i alo a zeo of f. Te Fudametal Teoem of Algeba Teoem If f (x) i a polyomial of degee wee > 0, te te equatio f (x) = 0 a at leat oe olutio i te et of complex umbe. Coollay If f (x) i a polyomial of degee wee > 0, te te equatio f (x) = 0 a exactly olutio povided eac olutio epeated twice i couted a olutio, eac olutio epeated tee time i couted a 3 olutio, ad o o. Te Complex Cojugate Teoem If f i a polyomial fuctio wit eal coefficiet, ad a + bi i a imagiay zeo of f, te a bi i alo a zeo of f. Decate Rule of Sig Let f (x) = a x + a x a x + a 1 x + a 0 be a polyomial fuctio wit eal coefficiet. Te umbe of poitive eal zeo of f i equal to te umbe of cage i ig of te coefficiet of f (x) o i le ta ti by a eve umbe. Te umbe of egative eal zeo of f i equal to te umbe of cage i te ig of te coefficiet of f ( x) o i le ta ti by a eve umbe. A98 Refeece

3 Fomula Algeba Slope m = y y 1 x x 1 Slope-itecept fom y = mx + b Poit-lope fom y y 1 = m(x x 1 ) Stadad fom of a quadatic fuctio f (x) = ax + bx + c, wee a 0 Itecept fom of a quadatic fuctio f (x) = a(x p)(x q), wee a 0 Stadad equatio of a cicle x + y = Expoetial gowt fuctio y = ab x, wee a 0 ad b > 1 Logaitm of y wit bae b log b y = x if ad oly if b x = y Vetex fom of a quadatic fuctio f (x) = a(x ) + k, wee a 0 Quadatic Fomula x = b ± b 4ac, wee a 0 a Stadad fom of a polyomial fuctio f (x) = a x + a 1 x a 1 x + a 0 Expoetial decay fuctio y = ab x, wee a 0 ad 0 < b < 1 Cage-of-bae fomula log c a = log b a, wee a, b, ad c ae poitive eal umbe log b c wit b 1 ad c 1. Sum of tem of 1 i = 1 1 = Sum of quae of fit poitive itege i ( + 1)( + 1) = i = 1 Sum of fit poitive umbe ( + 1) i = i = 1 Explicit ule fo a aitmetic equece a = a 1 + ( 1)d Refeece Sum of fit tem of a aitmetic eie S = ( a 1 + a ) Sum of fit tem of a geometic eie S = a 1 ( 1 1 ), wee 1 Recuive equatio fo a aitmetic equece a = a 1 + d Statitic Sample mea x = x z-scoe z = x μ σ Explicit ule fo a geometic equece a = a 1 1 Sum of a ifiite geometic eie a 1 S = 1 povided < 1 Recuive equatio fo a geometic equece a = a 1 Stadad deviatio (x σ = 1 μ) + (x μ) (x μ) Magi of eo fo ample popotio ± 1 Refeece A99

4 Tigoomety y Geeal defiitio of tigoometic fuctio Let θ be a agle i tadad poitio, ad let (x, y) be te poit wee te temial ide of θ iteect te cicle x + y =. Te ix tigoometic fuctio of θ ae defied a ow. (x, y) θ x i θ = y co θ = x ta θ = y x, x 0 cc θ = y, y 0 ec θ = x, x 0 cot θ = x y, y 0 Coveio betwee degee ad adia 180 = π adia Ac legt of a ecto Aea of a ecto = θ A = 1 θ Recipocal Idetitie 1 cc θ = i θ 1 ec θ = co θ Taget ad Cotaget Idetitie ta θ = i θ cot θ = co θ co θ i θ ecto θ 1 cot θ = ta θ ac legt 5π π 7π π 3π y π degee meaue 180 5π π π adia π meaue 3 π x 30 π 5π 4 3 π 11π 7π Pytagoea Idetitie i θ + co θ = ta θ = ec θ 1 + cot θ = cc θ Sum Fomula i(a + b) = i a co b + co a i b co(a + b) = co a co b i a i b ta a + ta b ta(a + b) = 1 ta a ta b Negative Agle Idetitie i( θ) = i θ co( θ) = co θ ta( θ) = ta θ Cofuctio Idetite i ( π θ ) = co θ co ( π θ ) = i θ ta ( π θ ) = cot θ Diffeece Fomula i(a b) = i a co b co a i b co (a b) = co a co b + i a i b ta a ta b ta(a b) = 1 + ta a ta b Pobability ad Combiatoic Numbe of favoable outcome Teoetical Pobability = Total umbe of outcome Pobability of te complemet of a evet P( A ) = 1 P(A) Pobability of depedet evet P(A ad B) = P(A) P(B A) Numbe of uccee Expeimetal Pobability = Numbe of tial Pobability of idepedet evet P(A ad B) = P(A) P(B) Pobability of compoud evet P(A o B) = P(A) + P(B) P(A ad B) Pemutatio P =! ( )! Combiatio C =! ( )!! Biomial expeimet P(k uccee) = C k p k (1 p) k Te Biomial Teoem (a + b) = C 0 a b 0 + C 1 a 1 b 1 + C a b C a 0 b, wee i a poitive itege. A100 Refeece

5 Peimete, Aea, ad Volume Fomula Squae Rectagle Tiagle w a c b P = 4 A = Cicle P = + w A = w Paallelogam P = a + b + c A = 1 b Tapezoid b d b b 1 C = πd o C = π A = π A = b A = 1 ( b 1 + b ) Rombu/Kite Regula -go d 1 d 1 a Refeece d d A = 1 d 1 d Pim Cylide A = 1 ap o A = 1 a Pyamid B P B P L = P S = B + P V = B L = π S = π + π V = π L = 1 P S = B + 1 P V = 1 3 B Coe L = π Spee S = 4π S = π + π V = 1 3 π V = 4 3 π3 Refeece A101

6 Ote Fomula Pytagoea Teoem a + b = c Simple Iteet I = Pt Ditace d = t a b c Compoud Iteet A = P ( 1 + ) t Cotiuouly Compouded Iteet A = Pe t Coveio U.S. Cutomay 1 foot = 1 ice 1 yad = 3 feet 1 mile = 580 feet 1 mile = 170 yad 1 ace = 43,50 quae feet 1 cup = 8 fluid ouce 1 pit = cup 1 quat = pit 1 gallo = 4 quat 1 gallo = 31 cubic ice 1 poud = 1 ouce 1 to = 000 poud U.S. Cutomay to Metic 1 ic =.54 cetimete 1 foot 0.3 mete 1 mile 1.1 kilomete 1 quat 0.95 lite 1 gallo 3.79 lite 1 cup 37 millilite 1 poud 0.45 kilogam 1 ouce 8.3 gam 1 gallo 3785 cubic cetimete Time 1 miute = 0 ecod 1 ou = 0 miute 1 ou = 300 ecod 1 yea = 5 week Tempeatue C = 5 (F 3) 9 F = 9 5 C + 3 Metic 1 cetimete = 10 millimete 1 mete = 100 cetimete 1 kilomete = 1000 mete 1 lite = 1000 millilite 1 kilolite = 1000 lite 1 millilite = 1 cubic cetimete 1 lite = 1000 cubic cetimete 1 cubic millimete = millilite 1 gam = 1000 milligam 1 kilogam = 1000 gam Metic to U.S. Cutomay 1 cetimete 0.39 ic 1 mete 3.8 feet 1 mete ice 1 kilomete 0. mile 1 lite 1.0 quat 1 lite 0. gallo 1 kilogam. poud 1 gam ouce 1 cubic mete 4 gallo A10 Refeece

Math III Final Exam Review. Name. Unit 1 Statistics. Definitions Population: Sample: Statistics: Parameter: Methods for Collecting Data Survey:

Math III Final Exam Review. Name. Unit 1 Statistics. Definitions Population: Sample: Statistics: Parameter: Methods for Collecting Data Survey: Math III Fial Exam Review Name Uit Statistics Defiitios Populatio: Sample: Statistics: Paamete: Methods fo Collectig Data Suvey: Obsevatioal Study: Expeimet: Samplig Methods Radom: Statified: Systematic: