RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify

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1 Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL EXPRESSION If a ad x are real umbers ad is a positive iteger, the x is a th root of a (ie x = a 1 = a ) if x = a 1) If a is positive ad is eve, the there exist TWO real th roots. Example 2 Solve for x a) x 2 = 16 b) x 2 = 11 c) x 4 = 81 d) x 4 = 5 2) If a is egative ad is eve, the there are NO real umber solutios. Example Solve for x a) x 2 = 25 b) x 4 = 7

2 ) If is odd, the there is ONE real th root of a. Example 4 Solve for x a) x = 8 b) x = 8 c) x 5 = 4 4) If a is zero, the there is ONE real th root of a, ad it is ZERO. Example 5 Solve for x a) x 2 = 0 b) x 5 = 0 Radical Properties from Math 10: 1) a 1 = a as discussed i above otes 2) a m 1 = (a ) m = ( a) m Example: 16 4 ) a m 1 = (a ) m = ( a) m = 1 Example: ) m 27 2 ( a 4) a b = a b Example: ) ab = a b Example: 12

3 1.2B Workig with Radicals absolute value Absolute Value is how may jumps the umber is from zero. Stated aother way, it is the distace from zero o the umber lie, regardless of directio. Distaces are always POSITIVE values. is jumps from zero, so the absolute value of, or =. is jumps from zero, so =. So if x =, x could have bee OR. For every absolute value solutio, there is a positive ad egative possibility. Example 1 - Evaluate a) 5 b) 7 c) 0.4 d) 5 6 e) 6 8 Example 2 - What are the possible values of x? a) x = 6 b) x = 9.7 c) x = 2 Radical Review Example - Idetify ad defie all parts of the radical, the simplify: 5 8 Roots of Positive Powers of x: Case 1: Whe x 0 i x with a positive iteger. The square roots of egative umbers are udefied i the set of real umbers. Therefore, if x 0, simplificatio is easier to realize. For example: x 2 = x 2 x = x x 4 = x 4 x 5 = x 5 x 6 = x 6 = = = = = Example 4 Simplify. Assume all variables represet positive umbers a) 16y 2 b) x 4 y c) 25x 5 y z 2 d) 8x 4 y 5 e) 27x y 6

4 Before Case 2 is discussed, it s importat to uderstad the Pricipal Square Root Theorem. Every positive umber has two square roots. Oe is positive, ad the other is egative. Example: x 2 = 16, so x = ± 16 = ±4 The PRINCIPAL SQUARE ROOT is the POSITIVE NUMBER SQUARE ROOT. Uless otherwise stated, the square root of a umber refers ONLY to the pricipal square root. Thus, 16 = 4 Case 2: Whe x is a real umber i x, with a positive iteger, we may require RESTRICTIONS to be placed o the expressio. If the expoet of a radicad is eve, the a NEGATIVE value will be chaged ito a POSITIVE value before the square root is take. Example: ( ) 2 = = = Whe variables exist i a radicad, it is ot kow if the variable represets a egative umber, zero, or a positive umber. A ABSOLUTE VALUE is sometimes eeded to esure that the result is a positive umber (required ONLY whe x chages from a EVEN power to a ODD power). Examples: x 2 = x x = x x 4 = x 2 (o absolute value eeded as x 2 ca oly result as a positive) x 6 = x etc. If the expoet i the radicad is ODD, the a NEGATIVE value of x will make the value NEGATIVE, which is UNDEFINED i the real umber system. Therefore, x 0 for all odd expoets. Examples: x = (x x) x = x x ; x 0 x 5 = x 7 = Example 5 Simplify. Let the variables be ay real umbers. a) 16x 2 b) 25x 2 y 4 c) x 14 d) 6x 2 y 5 Summary: 1) x is UNDEFINED (uless x 0) 2) x = a has o real solutio AND x = a has o real solutio ) x 2 = a has o real solutios, as does x 2 = a

5 1. Simplifyig Radicals Radicals ca be writte as fractioal expoets, as leared i Math 10. Examples: 2 = x = x 1 Geerally: a = a 1 Three Importat Relatioships of Radicals, all from Math 10: 1) a = a, a 0 because a = (a ) 1 2 = a Example: 5 2 = = 2) ab = a b, a, b 0 because ab = (ab) = a b = a b ) a b = a, a 0, b > 0 because a b b Simplifyig Expressios Cotaiig Radicals = ( a 1 a b )1 = Example 1 Simplify 20 Two Methods: 1 b = a b It is beeficial to kow the perfect squares up to 144, perfect cubes up to 125, ad perfect fourths up to 81. Example 2 Simplify a) 8 b) 27 c) 52 d) 24 e) f) 2

6 Example Simplify (assume variables are positive) a) 18x y 6 b) 6 7 p 4 c) 2x 8 y 11 d) 40a 4 b 8 c 15 e) 54a 5 b 10 4 f) m 7 4 g) 162x y 11 z 5 h) x1 64 Chagig Mixed Radicals to Etire Radicals Example 4 Chage to Etire (assume variables are positive) a) 4 b) 5 c) 2 7 d) 2x 6x e) x x f) a 2 b b 2 c g) x2 y 5 2xy 2

7 1.4 Addig ad Subtractig Radical Expressios Like Radicals Like Radicals work very similar to Like Terms. Simplify: x + 2x Simplify Like radicals have Steps for addig & subtractig like radicals: Example 1 - Simplify a) 7 2 b) c) d) e) f) 9b 16b, b 0 I example f, why does b have to be greater tha or equal to zero?

8 Example 2 Simplify a) 27xy + 8xy b) c) x 6y 5 28x 2 y, y 0 d) x4 y 5 xy 54xy 2

9 1.5A Multiplyig & Dividig Radical Expressios multiplyig radicals Example 1 - Multiply 2 5 ( 5) Verify your aswer: To multiply radicals: I geeral: Example 2 - Simplify: a) 5 ( 6) b) 2 6 (4 8) c) 2x (4 x) x 0 d) 2 11(4 2 ) e) (4 2 + )( ) f) ( x + y)( x 2 xy + y 2 )

10 dividig radicals Example - Divide Verify your aswer: To divide radicals: I geeral: Example 4 - Simplify: a) b) c) 18x x, x > 0 Multiplyig & Dividig Terms with Differet Idices Example 5 Simplify x ( x), x 0 Example 6 Simplify x, x > 0 x

11 1.5B Ratioalizig the Deomiator ratioalizig the deomiator A fial aswer caot have a radical i the deomiator. Therefore, you may have to ratioalize the deomiator a process that will elimiate the radical from the deomiator without chagig the value of the expressio. Example 1 - Ratioalize: a) 5 If the deomiator is a radical moomial, multiply the umerator ad deomiator by that radical. b) 2 7 c) d) e) 2 y f) 4 5x 4 g) 10x 2 x+1 Example 2 Ratioalize: a) If the deomiator is a radical biomial, multiply the umerator & deomiator by its cojugate. x 2 b)

12 c) a+ 2b a 2b, a, b 0 Example - Simplify a) 6, x > 0 b) 7, p 0 4x 2 9p Example 4 The surface area of a sphere is S = 4πr 2. If the surface area of the sphere is 144 mm 2, what is the radius?

13 1.6 Radical Equatios Radical Equatios are mathematical equatios that iclude a radical, such as: 2 6x 1 = 11. If the idex of the radical is eve, there are restrictios o the variable. Sice it is ot possible to fid the square root of a egative umber, the radicad caot be egative. Example 1 Fid the restrictio o the variable: a) 2 6x 1 = 11 b) x + 2 = 49 c) 7 2x + = 5 d) x + 4 = 2x 4 Steps to solvig radical equatios: 1. Fid the restrictios o the variable i the radicad (if the idex is eve). Remember, the radicad must be set to 0 ad the solved (if you multiply or divide by a egative umber to both sides, FLIP the iequality). 2. Get the radical all by itself o oe side of the equatio.. If the idex is 2, square both sides (if idex is, cube both sides, etc.) ad the solve for the variable. 4. See if the solutio if affected by the restrictio. 5. Check the aswer usig the origial equatio to see if solutios are valid or extraeous. Example 2 Solve a) 2 6x 1 = 11 b) 8 + y 5 = 2

14 Example Solve a) 4 + 2x = 1 b) 2 x 5 = 16 Example 4 - Solve a) 10x 7 = x b) 2 x = 7x + 6