CHAPTER 4 RADICAL EXPRESSIONS

Transcription

1 6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube root of 8 sce 8. Note that 8 has o other real cube roots. s a cube root of 8 sce ( ) 8. Note that 8 has o other real cube roots. s a fourth root of 6 sce 6. s also a fourth root of 6 sce ( ) 6. s a ffth root of sce. Note that has o other real ffth roots. s a ffth root of sce ( ). Note that has o other real ffth roots. a b. Note:. A real umber has two th roots whe s eve ad. oly oe th root whe s odd.. The odd th root of a postve umber s postve ad. the odd th root of a egatve umber s egatve. We use the otato b to deote the prcpal th root of a real umber b. The above otato s called a radcal expresso, b s called the radcad, s the dex of the radcal, ad the symbol dex s omtted whe the dex s. s called the radcal sg. The The prcpal root s chose to be the postve root for the case whe the dex s eve; t s chose to be the uque root whe s odd. For example, Note: b s udefed the set of real umbers whe b s egatve ad s eve. Thus, we say that 6 s udefed over the reals or s ot a real umber.

2 6. Smplfyg Radcal Expressos There are three codtos for a radcal expresso to be smplest form:. There are o perfect power factors uder a radcal of dex.. There ca be o fractos uder a radcal sg OR there ca be o radcals the deomator.. The dex should be the smallest possble. I ths secto we wll expla how to accomplsh the frst codto. We wll use the propertes ab a b for a a for b b where we assume that a ad b are such that the expressos do ot become udefed. Let us frst cosder umercal radcads. Examples s ot smplest form there s a perfect square factor 9 uder the radcal sg: OR s ot smplest form sce there s a perfect cube factor 8 uder the radcal sg: OR 8 80 s ot smplest form sce there s a perfect th power factor 6 uder the radcal sg: 80 6 OR 6 Next, let us cosder varable radcads. Let us also frst cosder the case whe the dex s odd. We observed Secto. that odd th roots follow the sg of the radcad, thus

3 6 For example, ad so o. x x for odd ad ay real umber x. x x, y y, m m, 7 7 Next, let us cosder the case whe the dex s eve. We defed Secto. the prcpal th root to be the postve root whe s eve. Cosder the followg: ( ) ad also ( ) ad also From the above we coclude that for ay real umber x ad eve dex, whch ca be abbrevated to x x f x s postve, ad x x f x s egatve x x for ay real umber x. Thus, for example: x x w w c c 6 6 If we wat to remove the cumbersome absolute value otato we eed to make the assumpto that all varables represet postve real umbers. Thus, wth ths assumpto we wll have x w c 6 6 x w c Let s ow smplfy radcals wth varable radcads. We wll assume that the varables ca be ay real umber. Examples. Smplfy: x x x x. Smplfy: 6 y y y 6

4 6 6. Smplfy: a 6 6 a a a 7. Smplfy: 8. Smplfy: x x x x x x x x 9. Smplfy: m m m m m m m 7 m 0. Smplfy: 8 p 8 p p p. Smplfy: w 6 w 6 w 9. Smplfy: 0 0. Smplfy: 6 x 8 6 x 8 x Now let us cosder a combato of umbers ad varable factors for radcads. We wll cotue to assume that the varables ca represet ay real umber.. Smplfy: 6 x y 6x y 6x y x y x y x y. Smplfy: a b c 0 6 8a b c 6a b c 6a b c 8 a b c a c b Smplfy: 0 p p 7 p p 7 p p p p 0 9 9

5 6 7. Smplfy: 6 7x y z 7x y z x y z x y x y z x y x y z x y Smplfy: 8 6 6a m e 6a m e 8a m e e 8a m e e a m e e a m m e Smplfy: 0 6y 6y 6y y 6y y y y y y Next, we have examples showg what to do whe there are factors outsde the radcal expresso. 0. Smplfy: ab 6a b 6 ab 6a b ab a b 0 ab ab a b 0 ab 6 ab ab ab 6a b ab. Smplfy: w 6y w w 6y w w 6y w w y w 8w y Fally, let us look at examples volvg ratoal radcal expressos. For these we wll assume that the varables represet ozero real umbers.. Smplfy: 6a b c 6 8 6a b 9a b 7 a b 7 a b 7 c c c c Smplfy: 0m 7 7 0m 8m m m m. Smplfy: 8x y xy z z 8 8 x y 6x y x y x y x y xy z z xy z z xy z z z z x y x y

6 66. Addg or Subtractg Radcal Expressos We ca add or subtract oly lke radcals,.e., those radcal expressos that have the same dex ad the same radcad. Examples Perform the operatos x y y x x y + y x x y y x x y + y x x y x y y x + y x x y + y x Recall that sometmes the radcals eed to be smplfed frst. I the followg assume that the varables represet postve real umbers y y y 8 8 y 9 y y y y y y y y y y y y y y 0y y. 6 8a + a 6a 6 8a + a 6a 6 8a 6a + a 6a 6 a 6a + a 6a a 6a + a 6a a 6a. 8 6 x y x xy x y x xy 6x xy x 8 xy x xy 6x xy 6x xy 8x xy x xy 6. a b ab a b a b ab a b a b a b ab a b b ( ) 0 ( ) a b b a b ab a b ab a b ab a b 6ab a b

7 67. Multplyg Radcal Expressos to multply radcals wth the same dex we use the followg: Examples. Multply: 0 a b ab Sometmes whe the umbers are large t s better to use prme factorzato rather tha carry out the multplcato.. Multply: Let us revew multplcato volvg varable radcads. We wll assume all varables represet postve real umbers.. Multply: x y xy x y xy x y xy x y x 6 y 6 y x y y Let us ow look at the case where we have both umercal ad varable factors.. Multply: ab a b ( ) ab a b ab a b 8a b 8a b a ab a 6 6 There could be factors sttg outsde the radcals.. Multply: ( ) ( y 8x y x x y ) ( ) ( ) y 8x y x x y y x 8x y x y 6xy x y 7 xy 6x y x xy xy x x y x 6 6 Oe or more of the factors could cosst of two or more terms. 6. Multply: ( ) ( )

8 68 7. Multply: ( + 8 )( 0 ) ( + 8 )( 0 ) ( 0 ) + ( ) + 8 ( 0 ) + 8 ( ) Multply: ( 6 + ) Oe way to do ths s to rewrte the problem as ( 6 + )( 6 + ) ad perform the multplcato as Example 7. Aother way s to apply the specal product formula ( ) a + b a + ab + b whch we wll do: ( 6 + ) ( 6 ) + ( 6 )( ) + ( ) Multply: ( 8 6 )( ) We ca multply as Example 7 but the aga we otce that we a b a b a b ca also use the specal product formula ( )( ) + as follows: ( 8 6 )( ) ( 8 ) ( 6 ) Multply: ( + 6 ) a + b a + a b + ab + b, Usg the specal product formula ( ) we get ( + 6 ) ( ) + ( ) ( 6 ) + ( )( 6 ) + ( 6 )

9 69 There could be varables the product. We aga assume that all varables represet postve real umbers.. Multply: ( x x + y )( x y y ) We apply FOIL. ( x x + y )( x y y ) x x x xy x y + y x y y y x 6x xy xy + xy y y x 6 xy xy + xy y The dex could be larger tha.. Multply: ( ) ( + ) OR Multply: ( ) x y xy x y ( ) x y xy x y 8x y 6x y xy x y 6 6. Multply: ( )( ) Note that the problem looks lke the specal product ( a b)( a + ab + b ) a b. ( 7 )( ) ( 7 ) ( 7 ) ( ) ( ) ( ) 7 7

10 70. Dvdg Radcal Expressos; Ratoalzg Deomators To dvde radcal expressos wth the same dex, we have the followg rule: a a, b 0 b b Ths meas that we smply dvde the radcads. Examples. Dvde: Dvde: Dvde: x 8y. Dvde: 8xy 6y x 8y x 8y x x x 8y 8y y y y x y 8xy 6y 8xy 6y y y y Multplcato ad dvso ca be combed: ab a b. ab 6 8 ab a b ab a b a b 6 a b ab ab ab ab 6 6

11 7 There could be two or more terms the dvded: 6. 8x y + 0 0xy xy 8x y + 0 0xy 8x y 0 0xy + x + y xy xy xy x + y x + 7y Note that radcals are NOT ALLOWED IN THE DENOMINATOR f a radcal expresso s to be cosdered smplest form. Let us ext cosder how the radcals ca be removed from the deomator. Ratoalzg the Deomator Ratoalzg the deomator meas makg the deomator a ratoal umber or removg radcals lke,,,... whch are rratoal umbers. Let us cosder the case where there s oly oe radcal ad oly oe term the dvsor. If the radcal s of dex, the dea s to multply ad dvde the expresso by a radcal that wll make the radcad a perfect power. Examples Ratoalze the deomator Smplfy: There could be varable factors as well (assume these are postve real umbers): 8 8. Smplfy: x 8 8 x 8 x 8 x x x x x x x x The dex could be larger tha : 9. Smplfy: Smplfy:

12 7 a. Smplfy: a a a a a a a a 8 a a a a a a Havg a fracto or ratoal expresso uder a radcal sg s the same as havg a radcal the deomator:. Smplfy: OR. Smplfy: xy x y 0x y xy xy x y xy xy Now let us cosder the case where there are two or more terms the deomator. For the case of dex radcals, we ca use the fact that ( )( ) a b a b a b + to remove the radcals from the deomator.. Smplfy:. Smplfy: 7 ( + 7 ) ( ) ( ) ( + ) ( + 7 ) ( ) + + ( ) ( ) ( ) ( ) ( ) ( ) or There could also be two terms the umerator:

13 7 6. Smplfy: ( ) There could be varables volved: 7. Smplfy: x x x + x x x x x x x x + x x + x x x 6x x x 9x x + 6x 6x x x + 6x 9x x ( x) ( x ) There could be radcals of dex : 8. Smplfy: + 7 As the case for dex radcals we ca use a specal product formula, amely ( )( ) factorg formula. a b a ab b a b + + +, whch we actually leared as a Smplfy: x x + x + x + x + a + ab + b so f we The deomator looks lke ( ) multply the umerator ad deomator by a b x ad apply the a b a ab b a b + +, we wll get formula ( )( ) ( ) ( ) ( ) x x x x x x. x + x + x x x 8

14 7.6 Ratoal Expoets I ths secto we wll lear how to mapulate expressos volvg ratoal or fractoal expoets. We defe a where a s a real umber ad s a teger wth as a a. The followg wll show how to use the above defto to smplfy expressos that look lke a. Examples. Evaluate: Note that ( ) ad ( ), thus the defto makes sese.. Evaluate: ( ) 8 ( ) Evaluate: Now let us cosder the case whe the umerator the ratoal expoet s ot. Let us look at how we ca hadle ths case: OR m m m a a a a m ( ) m m m ( ) a a a a Let us look at how these wll work the followg examples. m

15 7. Evaluate: OR ( ) ( ) 8 8 Note that the secod way seems better the sese that we deal wth smaller umbers.. Evaluate: 6 ( ) 6. Evaluate: Evaluate: ( ) 6 ( ) ( ) ( ) 8. Evaluate: 6 ( ) Evaluate: For ths problem t would make more sese to use the frst way of terpretg the ratoal expoet.. The expoet could be egatve: 0. Evaluate:. Evaluate: ( ) ( 8) ( ) 8. Evaluate: 8 ( 8) ( 8 ) ( ) ( 8)

16 76 8. Evaluate: Evaluate:. Evaluate: + ( ) ( ) ( ) ( ) Let us ext cosder expressos volvg varable factors. Aga let us assume all the varables represet postve real umbers. 6. Smplfy: ( ) 6 7x y ( ) 7x y 7x y xy Smplfy: ( ) 8 6 6a b a b 6a b a b a b 8a b ( ) ( ) ( ) 9 7m 8. Smplfy: 6 8p m 7m m 9m 6 6 8p 8p p p.7 Multplyg or Dvdg Radcals wth Dfferet Idces Now that we kow about ratoal expoets we ca talk about multplyg or dvdg radcals whose dces are ot the same. The dea s to wrte each radcal usg ratoal expoets. The goal s to frst make the dces the same. Sce the dex correspods to the deomator of the ratoal expoet, the dea s smlar to fdg the LCD for the expoets ad the covertg each expoet to ts equvalet fracto.

17 77 Examples. Multply: Multply: x x x x x x x x x x x x x x x x x x x x x x. Dvde:. Dvde: y y y y y y y y y y. Smplfy: x x x ( x) ( x) x x + ( x) ( x) ( x) ( x) 8x x ( x).8 Reducg the Idex The oto of ratoal expoets we ca expla why a aswer lke NOT cosdered to be smplest form. To see why ot, we have 6 6 x x x x ad thus the smplest form of x s 6 x s x, where we assume x s postve. Aother way to arrve at the same aswer s to use the followg: m a m a whch oe ca easly prove usg ratoal expoets. Thus, x x x. O the other had, 6 x x x. 6

18 78 Examples. Smplfy: OR ( ). Smplfy: x x x x x OR x x x 0 6. Smplfy: a b a b a b a b a b a b ( ) ( ) OR 0 0 a b a b a b.9 Complex Numbers I our prevous work wheever we ecoutered a expresso lke, 6,... we cocluded that the expresso s udefed (over the reals) or s ot a real umber. Whe we were solvg equatos elemetary algebra, wheever we reached a step, say, x 9, we cocluded that the equato has o real soluto. I ths secto we wll elarge the set of umbers that we are cosderg to clude umbers such as these umbers. We defe a complex umber to be ay umber of the form a + b, where a ad b are real umbers ad. a s called the real part of the complex umber ad b s called the magary part. +,, 0.. are examples of complex umbers. All the real umbers are complex umbers the magary part s 0 for a real umber. Whe the real part s 0, the complex umber s a pure magary umber. Two complex umbers a + b ad c + d are sad to be equal f ad oly f the real parts are equal ad the magary parts are equal,.e. a + b c + d a c ad b d. Addg or Subtractg Complex Numbers To add or subtract two complex umbers a + b ad c + d, we smply add/subtract the real part of oe to/from the real part of the other, the magary part of oe to/from the magary part of the other,.e. a + b + c + d a + c + b + d ( ) ( ) ( ) ( ) ( a + b ) ( c + d ) ( a c) + ( b d )

19 79 Examples. Add: 0 8 ad + ( ) ( ) Subtract 6 from +. ( ) ( ) Smplfy: ( ) + 9 ( ) Multplyg Complex Numbers To multply two complex umbers a + b ad c + d we smply apply the dstrbutve property ad remember that so that. ( )( ) ( ) ( ac bd ) + ( ad + bc) a + b c + d ac + ad + bc + bd ac + ad + bc bd It s ot ecessary to memorze the formula; oe ca smply perform the multplcato. It mght be terestg to ote some patter to make the multplcato faster: a + b Examples c +. Multply: ( )( + 7 ) ( )( ) d ( ) + ( + ) ac bd ad bc Expad: ( 6 ) ( ) ( ) ( )( ) ( ) Multply: ( )( 6 ) ( )( ) ( )( ) ( 6 0 ) ( 6 + ) Check what happes f you do ot do the secod step ad smply multpled drectly. Ca you expla why you got a slghtly dfferet aswer?

20 80 Dvdg Complex Numbers Suppose we wat to dvde the complex umber + by the complex umber. We ca wrte the problem as +. What do we do wth the above expresso? Note that sce f we rewrte the problem t wll look lke a expresso wth a radcal the deomator ad followg what we dscussed a prevous secto we ca remove from the deomator by followg a smlar procedure as ratoalzg the deomator. More precsely, we would eed the followg: The cojugate of a complex umber a + b s the complex umber a b. Let us gve a few examples: Note that the product of a complex umber ad ts cojugate s a real umber: ( )( ) ( ) a + b a b a b a b a + b To dvde a complex umber by aother complex umber, multply both the dvded ad the dvsor by the cojugate of the dvsor: Examples 7. Dvde: + a + b a + b c d c + d c + d c d Dvde: OR Complex Number 8 + ( ) Cojugate

21 8 Powers of We have the followg frst few powers of : Do you otce ay patter? Let us try the followg: 9. Evaluate: We ote above that, a postve teger,.e. rased to a postve multple of always gves the aswer. Thus, oe way of fgurg out what s: 0 There s a faster way but let us look at aother example frst: 0. Evaluate: Dd you observe that all we really eed s the remader whe the expoet s dvded by ad the we ca look at the frst colum of powers of I gve above to fd the aswer? Let s try the followg: Chapter Revew A real umber b s called the th root of a real umber a f For example, the th root of s sce a b.. We use the otato b to deote the prcpal th root of a real umber b. The above otato s called a radcal expresso, b s called the radcad, s the dex of the radcal, ad the symbol whe the dex s. s called the radcal sg. The dex s omtted The prcpal root s chose to be the postve root for the case whe the dex s eve; t s chose to be the uque root whe s odd. For example, 6 6 ad 7 8.

22 8 Smplfyg Radcal Expressos A radcal expresso to be smplest form f the followg are satsfed ( the examples we wll assume all varables represet postve real umbers):. There are o perfect power factors uder a radcal of dex. 8 x y x y x y x x 7a b 7a b b a b b There ca be o fractos uder a radcal sg or there ca be o radcals the deomator. x x x x x x x x a a 9a 9a a 7a a y y y y y y y y y y y y y y 0 y y 0 x x x + x x + x x x + x x x x + x x 9 9 a ab + b a ab + b a + b a + b a ab + b a + b. The dex should be the smallest possble. a b a b ab b ab 6 6 6x y z 6x y z xy z y z xyz 6 7 Addg or Subtractg Radcal Expressos We ca add or subtract oly lke radcals,.e., those radcal expressos that have the same dex ad the same radcad. Examples. x x 8 x + 6 x x 8 x 6x + 6 x x + x. b 0a b + 8a ab b a ab + 8a 9b ab b a ab + 8a b ab 6ab ab + ab ab 8ab ab

23 8 Multplyg Radcal Expressos A. For radcal expressos wth the same dex we use the followg: a b ab. Example xy x y 8x y xy y 7 B. For radcal expressos wth dfferet dces use ratoal expoets. 6 Example xy xy ( xy ) ( xy ) ( ) ( ) ( ) ( ) xy 6 xy 6 xy 6 xy x y x y x y y x y Apply the dstrbutve property ad/or specal product formulas whe multplyg radcals volvg two or more terms. Example ( ) ( ) ( ) a + b a + a b + b a + b a + b a + b a + b Oe ca also perform the expaso above by wrtg ( a + b) as ( a + b )( a + b ) ad applyg FOIL. Dvdg Radcal Expressos A. For radcal expressos wth the same dex use: a a, b 0 b b Example 80a b c 80a b c 6a b ab a bc a bc c c B. For radcal expressos wth dfferet dces use ratoal expoets. Example x ( x ) ( x ) x x x x x 8 x x x x x ( x ) 8 8 ( x ) Ratoalzg Deomators See codto for smplfed radcals.

24 8 Ratoal Expoets For a real umber a ad a teger wth, we have: Also, for a teger m, we have:. Examples Complex Numbers m ( ) m a a or a. ; ( ) ( ) ( ) m a a. We defe a complex umber to be ay umber of the form a + b, where a ad b are real umbers ad. We call a the real part of the complex umber ad we call b the magary part. + 8 s a complex umber; the real part s ad the magary part s 8. Operatos o Complex Numbers Examples. Add: ( ) ( ) ( ) ( ). Subtract: ( ) ( ). Multply: ( )( ) Expad: ( ) ( )( ) ( ). Dvde: Evaluate: ( )