Exponents and Radical

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Britney Wilkins
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1 Expoets d Rdil Rule : If the root is eve d iside the rdil is egtive, the the swer is o rel umber, meig tht If is eve d is egtive, the Beuse rel umber multiplied eve times by itself will be lwys positive Exmple 9 ot rel umber Exmple ot rel umber, 8 Rule : for y rel umber, the Therefore the priipl squre root of is the bsolute vlue of the root Exmple x x, Exmple 8x x Rule : If is whole umber tht is ot perfet squre the is irrtiol umber Exmple rtiol umber, Exmple irrtiol umber, 8 No rel umber No rel umber 8 8 No rel umber No rel umber No rel umber No rel umber 0 000, ,000 No rel umber x 8 x x 8x ( y + ) ( y + ) ( y + ) ( y + )

2 Rtiol Expoets Writig rtiol expoets s rdils or vie vers: The umertor beomes the power d the deomitor beomes the root. / ( ) m / m / Exmples m / 8 8 / ( ) / 8 / ( ) 9 9 A review of rules of Expoets / ( ) / 9 ( 9 ) m m+ m m b b m ( ) m ( ) m m m Prties b b As As / / / / / t t / / / 8 / / / / / / / ( ) / ( ) / 8x ( ) / b x b 9 ( ) / b 9 x / 8 8 /9 8 9 x / /0 9 9 x / 0 9 / 0 / b x / 8 x x

3 Rule to Multiply Rdils b b Rule to divide Rdils b b You multiply or divide the isides of rdil s log s they hve the sme roots. 8 b 0b 0 b b + b b b ( ) ( )( ) 8 Simplifyig Rdils You simplify terms iside the rdils, if ll terms re i produt form d their power is divisible by the root. Exmples b 8 bb b b b Prties x x b b b 0x x 0 8x y 80x x y x x x y x y ( x+ y) ( x+ y) xy x 0x xy x y x x 8y y 8 00y y / / 9 0 b y y y /+ / 9/ b b x y x y y x y y b ( ) x+ y x+ y b 9 b b b Addig d subtrtig rdils b You dd or subtrt if they hve the sme roots d rdids. Exmples + 0 x x x 8. 9 C ot be simplified

4 Prties Multiplyig d dividig rdils Exmples ( 8 + ) + + x x ( )( ) ( y y y ) + ( ) ( ) x y y y y Cojugte: Pir of rdil expressios tht the middle sig is differet d their produt will use rdils to be removed. x d + x d x x + x d x + re ojugte ( x )( x + ) x re ojugte ( + )( ) x x re ojugte ( )( ) x x x x + x x x x x x Usig ojugte llows us to rtiolize deomitors tht oti two terms. If there is oly oe term, o eed for ojugte, just multiply umertor d deomitor by deomitor multiply umertor d deomitor by ojugte of deomitor + ( + ) ( + ) ( + ) ( + )( 8+ )

5 Prties 0 + ( ) ( ) Solutio ( ) ( ) ( ) ( ) + Solvig Rdil Equtios ) Isolte oe of the rdils ) Rise both sides to the power of the roots of the rdil ) Simplify d solve ) Chek x x x ( x ) ( ) x + x No solutio beuse squre root of umber is ever egtive. + + ( x+ ) ( x+ ) x x x+ x+ x x x x + x 8 x x+ + x x x ( x ) ( ) x ( x ) ( ) + ( ) ( ) x 0x + x+ x x ( x 9)( x ) 0 x 9 x Prties x x + oly x 9 works Problem x + 8 x + 0 / x / x + 0 x x + x x

Pythgore theorem I right trigle, the formul tht desribe the reltioship betwee the hypoteuse d the two legs d b + b or + b Exmple : Fid the two legs of right trigle re d ihes wht is the legth of the

6 Pythgore theorem I right trigle, the formul tht desribe the reltioship betwee the hypoteuse d the two legs d b + b or + b Exmple : Fid the two legs of right trigle re d ihes wht is the legth of the hypoteuse? + diste betwee (,), (,), usig the diste formul d ( ) + ( ) 9 + Prtie : Fid the two legs of right trigle re d 8 ihes wht is the legth of the hypoteuse? Prtie : Use the Pythgore theorem to fid the height,( from the bse of the third floor to the groud). I this problem the legth of the ldder will be the hypoteuse. + h 0 h 0 00 h. Diste Formul The formul for the diste betwee two poits ( x, y), ( x, y) is give by Exmple : Fid the diste betwee (,), (,), usig the diste formul d ( ) + ( ) 9 + Prtie: Fid the diste betwee (, ), (8, ) Midpoit Formul d ( x x ) + ( y y ) The formul for the midpoit betwee two poits ( x, y), ( x, y) is give by M x + x y + y, + + Exmple : Fid midpoit betwee (,), (,), usig the diste formul M, (,0) Prtie: Fid the midpoit betwee (, ), (8, ).

7 i or i is lled imgiry umber or. Imgiry umber Exmple : ( ) i, Exmple : 8 8( ) ( ) i Complex umber A omplex umber is umber tht be expressed i the form + bi, where d b re rel umbers d i is the imgiry uit Exmple : + i, Exmple : 0+ i i Exmple : + 0i, Exmple : + i Add or subtrt Complex umbers Exmple : dd + i d i Exmple : dd + i d i + : ( + i) + ( i) + + i + : ( + i) ( i) + + i Multiply Complex umbers Exmple : ( + i)( + i) : ( )( ) ( + i) + i + i + i 8 + i i+ i 8 8i+ ( ) 8i Exmple : ( + i)( i) : ( + i)( i) i ( ) 9 ( i) Divide by Complex umbers If umber is divided by omplex umber, we eed to multiply tht the ojugte of its deomitor. Exmple : i, i i i i i i i Exmple :, + i ( i) ( i) ( i) ( i) i + i i 9i 9( ) + 9 Power of i Exmple : i ( i ) ( ) Exmple : i ( i ) ( ) ( i ) 8 8 Exmple : i i i ( i ) i ( ) i ( ) i i Exmple : ( ) ( ) ( ) + + i i i i i i i i i

RADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals

RADICALS. Upon completion, you should be able to. define the principal root of numbers. simplify radicals RADICALS m 1 RADICALS Upo completio, you should be ble to defie the pricipl root of umbers simplify rdicls perform dditio, subtrctio, multiplictio, d divisio of rdicls Mthemtics Divisio, IMSP, UPLB Defiitio: