P.5 Powers and Roots. Powers:

Transcription

1 P. Powers ad Roots. Powers: "Epoets" are a special otatio for repeated ultiplicatio. The will provide us with a shorthad for deotig epressios such as: Nael we will write this epressio as:.. I this eaple we call the "epoet" of the "ase ter " ad the "epoet" of the "ase ter." I other words a epoet sipl tells us how a ties the ase ust e repeated. Special Note: We will keep differet ases separate fro each other. Defiitio: For a positive iteger factors The uer is the ase ad the uer is the epoet. Rules of Epoets: The asic rules of epoets are: Rule : Eaple: That is whe ou ultipl like ases ou add the epoets. Rule : Eaple: That is whe ou divide like ases ou sutract the epoets.

2 Rule : ( ). Eaple: ( ) That is whe ou raise a power to a power ou ultipl the epoets. ********************************************************** Rule : Eaple: That is egative epoets result i "flippig fractios over." Rule : ( ) That is the th power of a product is equal to the product of the th powers of the factors. Eaple: ( ) 8 Note that a of these properties were give with ol two ters/factors ut the ca e eteded out to as a ters/factors as we eed. For eaple rule ca e eteded as follows. z z ( ) Warig: DO NOT distriute epoets across additio or sutractio! For eaple ( DOES NOT EQUAL! )

3 Rule : That is the th power of a quotiet is equal to the quotiet of the th powers. Eaple: Warig: DO NOT distriute epoets across additio or sutractio! For eaple DOES NOT EQUAL! *Zero epoets: Rule : 0 Eaple: ad ( ) 0 0 ************************************************************************ * Roots: If is a positive iteger that is greater tha ad a is a real uer the a a where is called the ide a is called the radicad ad the sol is called the radical. The left side of this equatio is ofte called the radical for ad the right side is ofte called the epoet for. Whe there is o ide uer it is uderstood to e a or square root. For eaple: square root of. CAUTION: If is eve ad a is egative the the root is ot a real uer. Rules: The asic rules which gover radical epressios are: Rule : a a Product Rule Note that if ou have differet ide uers ou CANNOT ultipl the together. Also ote that ou ca use this rule i either directio depedig o what our prole is askig ou to do.

4 Rule : a a Quotiet Rule This rule ca also work i either directio. Warig: Note that while we ca reak up products ad quotiets uder a radicalwe ca t do the sae thig for sus or differeces. I other words a a AND a a If we reak up the root ito the su (or differece) of the two pieces we clearl get differet aswers! So e careful to ot ake this ver coo istake! Eaples: Evaluate if possile or siplif each of the followig: a c. ( ) d. e. is udefied; does ot eist. We are goig to e siplifig ore radicals so we should et defie siplified radical for. A radical is said to e i siplified radical for (or just siplified for) if each of the followig are true.. All epoets i the radicad ust e less tha the ide.. A epoets i the radicad ca have o factors i coo with the ide.. No radicals appear i the deoiator of a fractio. Eaples: Siplif each of the followig. () z (c) 8 (d) Solutio:

5 To siplif the give prole we eed to reak dow the radicad ito perfect squares sice the ide is. Appl the product rule the siplif. So () z To siplif the give prole we eed to reak dow the radicad ito perfect rth power factors sice the ide is. The appl the product rule. So z z. z. z... z. Now tr these. (c) 8 (d) Eaples: Siplif each of the followig. () (c) a (d) c d Solutio: To siplif the give prole we eed to reak dow the radicad ito perfect squares sice the ide is. Appl the quotiet rule the siplif. So () (c) a (d) c d

6 More Practice proles:. Siplif the followig. Epress aswers i ters of positive epoets. ( a ) () (c) (d) (e) z (f) z (g) ( ) ( ) (h) ( )( ) (i) a c (j) 0 ( )( ) (k) 0 ( ) (l). Siplif the followig radicals: 8a z () (c) 8 z 8 0 a (d) c d (e) 0 a (f) 8 8 (g) Please check hoework proles as well.

7 P. Fractioal Epressios:. Operatios with Ratioal Epressios: * Equivalet fractios: * Ratioal Epressios: A ratioal epressio is the quotiet of two poloials. E: * Siplifig Ratioal Epressios: NOTE: To siplif a ratioal epressio factor copletel the uerator ad deoiator the cacel coo factors. A ratioal epressio is reduced to lowest ters if all coo factors fro the uerator ad deoiator are caceled. Eaple a: Reduce to lowest ters Not reduced to lowest ters ( )( ) ( )( ) With ratioal epressios it works eactl the sae wa. reduced to lowest ters Not reduced to lowest ters ( )( ) ( ) reduced to lowest ters ** We have to e careful with cacelig. There are soe coo istakes that studets ofte ake with these proles. Reeer that i order to cacel a factor it ust ultipl the whole uerator ad the whole deoiator. So the aove could cacel sice it ultiplied the whole uerator ad the whole deoiator. However the s i the reduced for ca ot e cacelled sice the i the uerator is ot ties the whole uerator. To see wh the s do t cacel i the reduced for aove put a uer i ad see what happes. Let s plug i. (If gets caceled) Clearl the two aswers are ot the sae uer! Note: Ol COMMON FACTORS of the uerator ad deoiator ca e caceled.

8 Eaple : Siplif the followig: a 8 a a az a a z 8 a a * Multiplig Ratioal Epressios: ) Copletel factor each uerator ad deoiator. ) Multipl the uerators ad ultipl the deoiators. ) Siplif the result as far as possile cacellig coo factors. Eaple : ( ) ( )( ) ( ) ( ) ( )( ) ( ) Eaple : Dividig Ratioal Epressios: ) Copletel factor each uerator ad deoiator. ) Chage to ultiplicatio. ) Ivert (flip) the secod fractio ad proceed as i ultiplicatio. Eaple : ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( )( ) 8

9 * Fidig the Least Coo Deoiator (LCD): ) Copletel factor each deoiator. ) The LCD is the product of all uique factors each raised to the greatest power that appears i a factored deoiator. Eaple : ) z ) z z z ) 0 ) Hit: If opposite factors occur do ot use oth i the LCD. Istead factor - fro oe of the opposite factors so that the factors are the idetical. E: If ou have factors like - ad - these are called opposite factors. Notice that ou ca factor a - fro - so that the factors are idetical. Addig or Sutractig Ratioal Epressios: ) With the sae deoiator: If the deoiators are the sae keep the sae deoiator just add uerators. The siplif if possile. Eaple : ( ) ( )( ) Eaple : ( )( )

10 ) With differet deoiators: (LCD is NEEDED) a. Factor copletel each deoiator.. Fid the LCD of the ratioal epressio. c. Write each ratioal epressio as a equivalet ratioal epressio whose deoiator is the LCD foud i step () d. Add or sutract uerators ad write the result over the coo deoiator. e. Siplif the resultig ratioal epressio if possile. Eaple 8: a) ) c) d) e) * Cople Fractios: A cople fractio is a ratioal epressio whose uerator deoiator or oth cotai oe or ore ratioal epressios. Eaples are a The ca e siplified treatig the uerator ad the deoiator as separate proles. The we have a "divisio" prole. 0

11 For eaple to siplif we first coplete the sutractio prole cotaied i the uerator of the etire fractio:. Now we have the divisio prole: We ivert ad ultipl:. As efore we should ow factor i order to reduce: ( )( ) Now cacel the coo factor " ( )." ( ) So our fial aswer i factored for reduced to lowest ters is:. Reeer ou do ot eed a coo deoiator whe ultiplig (or dividig) fractios. Steps for siplifig cople fractios: ) Siplif the uerator ad the deoiator of the cople fractio so that each is a sigle fractio. ) Perfor the idicated divisio ultiplig the uerator of the cople fractio the reciprocal of the deoiator of the cople fractio. ) Siplif if possile.

12 Eaple : Siplif the followig: a) ) c) d) Solutio: a) ( ) ) First we siplif the uerator ad deoiator separatel so that each is a sigle fractio Note the LCD of the fractios that are i the uerator is ad the LCD of the fractios that are i the deoiator is.

13 Now tr the followig proles:. Perfor the idicated operatios ad siplif our aswers. () (c) (d) 0 (e) (f) (g) (h) h h (i) a a