Radical Epressions Wha are Radical Epressions? A radical epression is an algebraic epression ha conains a radical. The following are eamples of radical epressions + a Terminology: A radical will have he following; a radical sign, a radicand, and an inde. Inde Radical Sign n a Radicand Square Roo: If he inde of a radical is hen he radical is called a square roo and he inde of is no wrien. The following epressions are square roos. Definiion of a Square Roo: The number b is a square roo of a if a b = Eample: a = b because b = a. = because =. = because =. =. because q = q because = a for any real numbers a and b and a is nonnegaive. ( q ) = q Perfec Squares: All of he radicands in he above eample were perfec squares. Tha means ha here is a value ha when squared will resul in he radicand. This value is called he perfec square roo. I will be very helpful o know he perfec squares of he numbers 9 = = = 9 = = 9 0 = = 9 = = = 00 = = = 9 = 9 =
Knowing hese perfec squares will help you o find he perfec square roos. I is imporan o undersand he relaionship beween perfec squares and heir square roos. As an eample, 9 is he square of and is he perfec square roo of 9. Likewise, is he square of and is he perfec square roo of. Eample: Find he square roo of each of he following:. 9. Soluion: Since hese are all perfec squares hey have perfec square roos. = 9. 9 =. = A variable is a perfec square if i is raised o an even power. Therefore, any square roo wih a radicand conaining variables raised o an even power will be a perfec square and consequenly have a perfec square roo. The square roo will be he variable raised o a power which is he original power divided by. Eample: Find he square roo of each of he following.... q r s 0 Soluion: Since each radicand conains variables raised o even powers, hey are all perfec squares. The square roos of each will be he variables raised o a power equal o he original power divided by. = =... = = q = q = 0 q 0 r s = r s = rs
Eample: Find he square roo of each... 9 y w. Soluion: =. 9 y = y. w = 9w. No real soluion. The radicand canno be negaive Cube Roos: If he inde of a radical is hen he radical is called a cube roo. In his case he inde is is and is wrien. The following epressions are cube roos. a Definiion of a Cube Roo: The number b is a cube roo of a if negaive. b = a for any real numbers a and b. Wih cube roos he radicand may be Eample: a = b because b = a. = because =. = because =. = because =. 9 q = q because 9 ( q ) = q. = because ( ) =
Perfec Cubes: All of he radicands in he above eample were perfec cubes. Tha means ha here is a value ha when cubed will resul in he radicand. This value is called he perfec cube roo. I will be very helpful o know he perfec cubes of he numbers. = = = = = ( ) ( ) ( ) ( ) ( ) = = = = = Knowing hese perfec cubes will help you o find he perfec cube roos. I is imporan o undersand he relaionship beween perfec cubes and heir cube roos. As an eample, is he cube of and herefore, is he perfec cube roo of. Likewise, is he cube of and herefore is he perfec cube roo of Eample: Find he cube roo of each of he following:.. Soluion: Since hese are all perfec cubes hey have perfec cube roos. =. =. = A variable is a perfec cube if i is raised o a power which is a muliple of. Therefore, any cube roo wih a radicand conaining variables raised o a power divisible by will be a perfec cube and consequenly have a perfec cube roo. The cube roo will be he variable raised o a power which is he original power divided by. Eample: Find he cube roo of each of he following.... 9 r s w
Soluion: Since each radicand conains variables raised o powers ha are divisible by, hey are all perfec cubes. The cube roos of each will be he variables raised o a power equal o he original power divided by.... = = = = 9 9 = = r s w = r s w = rs w Eample: Find he cube roo of each. 9. y d e f. Soluion:. = 9 y = y. d e f = d e f Nh Roos: Jus as we can raise a number o a power oher han or, we can also find roos of a number oher han he square roo and he cube roo. Eample: The following are eamples of nh roos. h roo n =. h roo n =. y h roo n =. h roo n =
Definiion of an nh Roo: The number b is he nh roo of a if n > b n = a, for any real number a and for any posiive ineger Eample: n a = b because b n = a. = because =. = because =. = because =. q = q because ( q ) = q The menal process for finding nh roos is similar o finding square roos or cube roos. Eample: Find he nh roo of he following... 0 r s.. Soluion: y q 0 0 = =. r s = r s = rs. = =. y = y = y. q = q = q
.-Applicaions: Eample: If an objec is dropped, he ime i akes in seconds for he objec o fall s fee is given by he epression s If a sone is dropped from a heigh of 9 fee above he ground, find he ime i akes o hi he ground. Soluion: Subsiue s = 9 ino he epression and simplify. s = 9 = = I will ake he sone seconds o hi he ground. Eample: The lengh of he side of a square wih an area A can be compued using he epression A. The area of he square picure, including he -inch frame is 9in. Find he lengh of he side of he picure frame and he sie of he phoograph in he frame. Soluion: Begin by subsiuing he area of he frame ino he epression A. = A = The lengh of he picure frame is inches. Since he border is -inch, he sie of he phoograph will be -=. 9 Eample: The geomeric mean is a saisic used in business and economics. The geomeric mean of hree numbers is given by he epression p, where p is he produc of he numbers. Find he geomeric mean of 9, and. Soluion: The produc of 9,, and is Subsiue his value ino he epression for p. p = = Therefore, he geomeric mean of, 9, and is.