Tallahassee Community College. Simplifying Radicals


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1 Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is the numer whose squre root is eing found. 8 euse euse The smol is lled rdil nd it is red s the squre root of. The numer underneth the rdil is lled the rdind. In the epression., the rdind is It should e noted tht eh positive numer hs squre roots. One is the positive or prinipl squre root, nd the other is the negtive squre root. euse euse ( ) We re usull interested in the positive squre root. If we wnt the negtive root, we put negtive sign in front of the rdil. 7 Note tht we nnot hve negtive sign under the rdil. is not rel numer, euse there is no numer tht we n multipl itself nd get. (In MAT 10 nd MAC 110 ou will lern how to del with this sitution.)
2 To simplif rdil we must look for nd remove n perfet squre ftors tht m e in the rdind. REMEMBER tht perfet squre is the squre of n integer. 1 Perfet squres 8 A rdil epression is in simplest form if the rdind ontins no perfet squre ftors. To simplif rdil we will first find the prime ftoriztion of the rdind nd rewrite the rdind in eponentil form. prime ftoriztion of in eponentil form. If the eponent is n even numer, then the numer itself is perfet squre. To tke the squre root of EXAMPLE: we remove the rdil nd divide the eponent. / 8 One we hve divided the eponent, we n multipl out the remining ftors. EXAMPLES: 81 / 1 / / / 7 / 7 1 Often the numer we wish to simplif is not perfet squre. We then hve to find n perfet squre ftors ontined in the numer nd remove them from under the rdil tking their squre roots. To simplif 0, first find the prime ftoriztion of 0. 0 NOTICE tht the eponents re odd numers. This mens tht nd 1 re not perfet squres.
3 An prime ftor with n eponent of 1 will not e perfet squre nor will it ontin perfet squre. An prime ftor with n even eponent will e perfet squre. An prime ftor with n odd eponent of or higher will ontin perfet squre ftor ontins perfet squre is written s 1 The Produt Propert of Squre Roots llows us to rewrite produt under rdil s produt of seprte rdils We now hve on rdil whih is perfet squre nd one whih is not. We n tke the squre root of the perfet squre nd multipl the ftors remining under the rdil. 10 The omplete proess is s follows: Find the prime ftoriztion of 0 s Seprte the perfet squres Tke squre roots. The solution is red s times the squre root of 10 EXAMPLE: Simplif / 1 1 Find the prime ftoriztion of s Seprte the perfet squres Tke squre roots. Simplif to get The solution is red s times the squre root of
4 EXAMPLE: Simplif 180. Notie tht this is times the squre root of 180. We must simplif 180 first nd then multipl. 180 Find the prime ftoriztion of 180 Seprte the perfet squres / / Tke squre roots. Multipl 0 The solution is red s 0 times the squre root of EXAMPLE: Simplif / Find the prime ftoriztion of s Seprte the perfet squres Tke squre roots. Simplif to get nd multipl The solution is red s times the squre root of Mn of the epressions we will need to simplif will ontin vriles. euse ( ) euse ( ) An vrile rdil epression whih hs n even eponent will e perfet squre. An vrile rdil epression whih hs n eponent of 1 will not e perfet squre nor will it ontin perfet squre ftor. An vrile epression whih hs n odd eponent of or higher will ontin perfet squre ftor. /
5 EXAMPLES: / 18 / 18 8 / 8 We often hve rdils whih hve oth numers nd vriles. EXAMPLE: Simplif EXAMPLE: Simplif EXAMPLE: Simplif 7 7 Find the prime ftoriztion of s Seprte the perfet squres nd tke squre roots Find the prime ftoriztion of 7,, nd Seprte the perfet squres nd tke squre roots Multipl the numers Find the prime ftoriztion of 7 nd Seprte the perfet squres nd tke squre roots Multipl the numers nd multipl the vriles
6 EXAMPLE: Simplif Find the prime ftoriztion of 18 s Seprte the perfet squres nd tke squre roots Multipl the numers nd multipl the vriles EXERCISES. Simplif eh of the following. ) ) 1 ) 0 d) 80 e) 1 f) 1 g) h) 0 i) 8 10 j) 10 KEY ) 7 ) ) 10 d) 1 e) g) j) 0 7 f) 7 h) i) 1
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