Section 6.1 Radicals and Rational Exponents


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1 Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig the symbol. Progress Check 1 Fid the square roots of 49. Progress Check Fid each square root that is a real umber. Approximate irratioal umbers to the earest hudredth, ad idetify all umbers that are ot real umbers. 00 b. 5 c. 5 d. 5 Defiitio of th Root For ay positive iteger, the umber b is a th root of a if b Progress Check 3 Fid each root that is a real umber b. 15 c d. 3 7 Defiitio of a 1/ If is a positive iteger ad a is a real umber, the Progress Check 5 Evaluate each expressio b a. a 1
2 Ratioal Expoet If m ad are itegers with > 0, ad if m/ represets a reduced fractio such that real umber, the Progress Check 6 Evaluate each expressio. 5 4 b. 5 m m/ m a a a 7 4 c a is a Progress Check 7 Evaluate each expressio b. 3 Progress Check 7 Perform the idicated operatios, ad write the result with oly positive expoets. Assume all variables represet positive real umbers b. x x c. 3 9y d. 1 3 e. a a b b f. 3y 4 4 y 3
3 Sectio 6. Product ad Quotiet Properties of Radicals For real umbers a, b, a, ad b : 1. a b a b a a. ( 0) b b b Progress Check 1 Simplify each radical. 75 b. 80 c. 3 3 Note: a a ad a a Example: Progress Check Simplify each radical. Assume x0, y b. 4 1 y c d. 9 6 x Progress Check 3 Simplify each radical. Assume x0, y x b x y * ot part c. from text x y Progress Check 6 Multiply ad simplify each radical. Assume x0, y c. 4x y 6xy * ot part of this PC xy 5x y 3
4 Progress Check 4 Simplify each radical. Assume y b c. 8 y Progress Check 7 Divide ad simplify where possible. Assume y 0. b. 7 Progress Check 8 Ratioalize each deomiator. Assume x0, y 0. 1 x 8 b. c. y 1 Simplified Radical To write a radical i simplified form: 1. Remove all factors of the radicad whose idicated root ca be take exactly.. Write the radical so that the idex is as small as possible. 3. Elimiate all fractios i the radicad ad all radicals i the deomiator. (which is called ratioalizig the deomiator) 4
5 Sectio 6.3 Additio ad Subtractio of Radicals To add or subtract radicals, the radicals have to be like. Like radicals are radicals that have the same radicad ad the same idex. Oly like radicals ca be combied. Progress Check 1 Simplify where possible. Assume x0, y b. 8 c. x x d. x y x y e. y x x y Progress Check Simplify where possible. Assume x0, y c. y 63x y x 8y 3 5
6 Sectio 6.4 Further Radical Simplificatio Progress Check 1 Simplify each radical. 3 b x c d. 6 10y Progress Check Simplify each expressio b Progress Check 3 Simplify each expressio. Assume x b. 4 x4 x Progress Check 3 Simplify each expressio. Assume x b. x 3 x 6
7 Sectio 6.5 Radical Equatios To Solve Radical Equatios 1. If ecessary, isolate a radical term o oe side of the equatio.. Raise both sides of the equatio to a power that matches the idex of the isolated radical. 3. Solve the resultig equatio, ad check all solutios i the origial equatio. Progress Check Solve 3x 4 3. Priciple of Powers If P ad Q are algebraic expressios, the the solutio set of the equatio P = Q is a subset of the solutio set of P Q for ay positive iteger. Solutios of P solutios. Progress Check 4 Solve 0 8x x. Q that do ot satisfy the origial equatio are called extraeous Progress Check 5 Solve 1 x11 x. Progress Check 6 Solve 4x1 x 1. 7
8 Sectio 6.6 Complex Numbers To defie a complex umber, we first itroduce a ew set of umbers called imagiary umbers i which square roots of egative umbers are defied. The basic uit i imagiary umbers is 1, ad is desigated by i. Thus, i 1 ad i 1. Priciple Square Root of a Negative Number If a is a positive umber, the a i a. Progress Check 1 Express each umber i terms of i: a. 9 b. 7 c. 9 d. 8 Defiitio of a Complex Number A umber of the form a + bi, where a ad b are real umbers ad i 1, is called a complex umber. The umber a is called the real part of a + bi, ad b is called the imagiary part of a + bi. Addig ad Subtractig Complex Numbers Progress Check Combie the complex umbers: 9 5 b. ( 1 i) (8 5 i) c. (9 3 i) (16 i) Multiplyig Complex Numbers Example 3 Multiply the complex umbers: 5 4 b. (3 4i)( 6i) c. (5 i) 8
Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
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