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Name Class 6-1 Date Notes Roots and Radical Expressions For any real numbers a and b and any positive integer n, if a raised to the nth power equals b, then a is an nth root of b. Use the radical sign to write a root. The following expressions are equivalent: What are the real-number roots of each radical expression? a. 3 343 Because (7)3 343 , 7 is a third (cube) root of 343. Therefore, 3 343 7 . (Notice that (7)3 343 , so −7 is not a cube root of 343.) 4 4 1 b. 4 625 1 1 1 1 1 1 Because and , both and are real-number 625 5 5 5 5 625 1 fourth roots of . 625 c. 3 0.064 3 Because ( 0.4) 0.064 , −0.4 is a cube root of −0.064 and is, in fact, the only one. So, 3 0.064 0.4 . d. 25 Because (5)2 = (−5)2 = 25, neither 5 nor −5 are second (square) roots of −25. There are no real-number square roots of –25. Exercises Find the real-number roots of each radical expression. 1. 169 2. 3 729 3. 4 0.0016 1 4. 3 8 5. 4 121 125 6. 3 216 Name Class 6-1 Date Notes (continued) Roots and Radical Expressions You cannot assume that n a n a . For example, (6)2 36 6 , not −6. This leads to the following property for any real number a: n an a If n is odd n an a If n is even What is the simplified form of each radical expression? a. 3 1000x3 y9 3 1000 x3 y9 3 103 x3 ( y3 )3 b. 4 256 g 8 h 4 k16 Write each factor as a cube. 3 (10 xy3 )3 Write as the cube of a product. 10xy3 Simplify. 8 4 2 4 4 256 g 4 4 ( g ) Write each factor as a power of 4. h4k16 h 4 (k 4 )4 4g 2 4 hk 4 4g 2 h k4 Write as the fourth power of a quotient. Simplify. The absolute value symbols are needed to ensure the root is positive when h is negative. Note that 4g2 and k4 are never negative. Exercises Simplify each radical expression. Use absolute value symbols when needed. 1 7. 36x 2 8. 3 216y3 9. 100x 2 10. x 20 y8 11. 3 ( x 3)3 ( x 4)6 12. 5 x10 y15 z5