Precalc0001to0005-page25

Precalc0001to0005-page25 - n is even, we require x . 13 x 2...

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(Section 0.5: Exponents and Radicals: Laws and Forms) 0.5.3 Laws of Radicals If the expressions involved are real -valued, then the following laws apply. • Law 13 fundamentally distinguishes between even and odd roots. • For the others, the square root laws extend to even roots under nonnegativity constraints and odd roots without such constraints. For example, xy 3 = x 3 () y 3 () for all real x and y . # Law In Plain English / Comments 9 xy = x y , if x ± 0 and y ± 0 The root of a product equals the product of the roots . See Law 4, with n = 1 2 , 1 3 , etc. 10 x y = x y , if x ± 0 and y > 0 The root of a quotient equals the quotient of the roots . See Law 5, with n = 1 2 , 1 3 , etc. 11 x m n = x mn (if m or n is even, we require x ± 0 ) For example, x 3 = x 6 , if x ± 0 . This is related to Law 3, with m and n there being the reciprocals of m and n here. 12 x () 2 = x , if x ± 0 More generally, x n () n = x , if n = 2, 3, 4, . If
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Unformatted text preview: n is even, we require x . 13 x 2 = x x 3 3 = x x n n = x , if n = 2, 4, 6, See Warning 4 . ( ) x n n = x , if n = 3, 5, 7, For example, we simplify 18 using Law 9: 18 = 9 2 = 9 2 = 3 2 . (The greatest perfect square that divides 18 is 9.) WARNING 2 : Although Laws 9 and 10 cover the square root of a product or a quotient , there is no similar law for the square root of a sum or a difference . (See the Exercises.) WARNING 3 : Do not apply Laws 9-12 to even roots if x < or y < ; we must then deal with imaginary numbers, which are not real numbers. As we will see in Section 2.1, 2 3 6 . WARNING 4 : See Law 13. The statement x 2 = x is incorrect if x < 0. For example, if x = 3 , 3 ( ) 2 = 9 = 3 , not 3 ....
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