8.2 Simplifying Radicals
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- Derrick Wheeler
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1 . Simplifig Rdicls I the lst sectio we sw tht sice. However, otice tht (-). So hs two differet squre roots. Becuse of this we eed to defie wht we cll the pricipl squre root so tht we c distiguish which oe we wt. Defiitio: Pricipl th root The pricipl th root of umer is the th root tht hs the sme sig s the origil umer. We use rdicl ottio to idicte we the pricipl th root. So this defiitio, sice the is positive d sice the is egtive. The ide is this, If ou hve eve ide, the pricipl th root must e positive (sice if is eve, the rdicd must e positive or else ou do t get rel umer) If ou hve odd ide, the pricipl th root must hve the sme sig s the rdicd Now otice the followig ( ) 9 d 9 Recll for ( ) we would hve to follow the order of opertios which mes evlutig the squre first. The reltioship illustrted ove motivtes the followig propert. Propert If is eve, the If is odd, the. For emple, ( ) Ad.. For emple, ( ). I order to simplif the mtter, we will lws ssume tht the vriles re positive. Therefore we will ot eed to worr out the solute vlue sigs. I tht cse the first prt of the propert ecomes: If is eve, the. So, wht this tells us is wheever the ide d the power i the rdicd mtch, ou ed up with just the prt iside. Tht is to s, Whe the re the sme, the epoet d the rdicl ccel. We use this ide with our rules for rtiol epoets to simplif rdicls. We illustrte with the followig emple. Emple : Simplif... c. c d. 0 Solutio:
2 . First otice tht the egtive is outside the rdicl d therefore does ot cuse prolem. The prolems ol rise whe we hve egtive uder eve ideed rdicl. So we c simpl covert the rdicl ito rtiol epoets d the simplif s efore s follows ( ).. Agi, lets covert the rdicl ottio ito epoet ottio d simplif ccordigl. ( ) c. Likewise we simplif covertig d simplifig. c ( c ) c d. This time we eed to just e creful out the umericl prt of the rdicd. We del with tht the sme w we lws del with it. We cotiue s follows. 0 0 ( ) ( ) Recll tht it is ok to hve egtive uder odd ide root. It simpl mes tht the umer is egtive. This emple illustrted how to simplif rdicl if the powers udereth re ice powers. However, we kow tht sometimes the powers re ot ice. So we eed to e le to simplif rdicl o mtter how complicted. For this we eed the followig. A rdicl is i simplest form whe:. The rdicd hs o fctors tht hve power greter th the ide.. No frctios re udereth the rdicl.. No rdicls re i the deomitor. I the rest of this sectio we wt to cocetrte o the first oe of these rules. Tht is, the rdicd hs o fctors tht hve power greter th the ide. The other two rules we will del with lter. I order to del with prt oe of the rule we will eed the followig propert. Product Propert of Rdicls If d re rel umers the, To illustrte this cosider. We kow tht this is. However, we should e le to get eve if we use the propert. We c do so s follows 9 9
3 This is o mes proof of the propert, merel emple to illustrte its vlidit. As we sid, we use this propert to help simplif rdicls. Emple : Simplif c. d. 0 Solutio:. I light of emple, we should tr to fid w to mke the powers uder the rdicl ecome multiples of the ide. We kow the tht the would esil simplif. Sice there is o ide show, it is two. So lets rewrite ech vrile s hvig power tht is multiple of two, times whtever else we eed. We do so s follows 9 Now we c group together ll of the prts tht hve the ice powers i the frot d ll the etr stuff i the ck. Usig the product propert for rdicls we get ( ) ( ) Notice tht we c ow simplif the frot prt s we did i emple. ( ) All the powers uder the rdicl re smller th the ide d so the rdicl is simplified.. We see the tht the oject is to write the rdicd s hvig powers tht re multiples of the ide times whtever is left over. This is the ke to simplifig rdicls. However, for this emple, we hve to del with the 0 s well. To do tht we will rek the 0 dow ito its prime fctoritio d the use the sme techique s we did with the vrile prts, tht is, write s power tht re multiples of the ide. So, the prime fctoritio of 0 is. So we hve 0 Sice the ide is here, we will mke the powers multiples of d put the left over prts together t the ck of the rdicl d proceed like ove. This gives us ( ) ( ) Notice, ll powers re smller th the ide. Therefore, the rdicl is simplified. c. Agi we will strt prime fctorig the :. Now we cotiue s efore, tht is, mke everthig i the rdicd hve power tht is multiple of the ide times the left over stuff. We proceed s follows
4 0 9 ( ) ( ) 9 9 Notice, sice the ide is odd, the egtive uder the rdicl c just e crried out sice we kow tht swer will e egtive. d. Lstl, we proceed s we hve for ll the other emples. 0 0 ( ) ( ) So gi, the ke to the first prt of simplifig rdicls is to rewrite the powers uder the rdicl s multiples of the ide. The we simpl eed to use the product propert d properties of rtiol epoets to fiish.. Eercises Simplif. Assume ll vriles represet positive vlues c c. 0 c
5 c c c. 0
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