A88 APPENDIX Review 2 Simplify Radicals Let n 2 ≥ and m 2 ≥ denote integers, and let a and b represent real numbers. Assuming that all radicals are defined, we have the following properties: Using a Calculator to Approximate Roots Use a calculator to approximate 16. 5 Solution Figure 39 shows the result using a TI-84 Plus CE graphing calculator. EXAMPLE 2 Properties of Radicals ab a b n n n = (2a) a b a b b 0 n n n = ≠ (2b) a a n m n m ( ) = (2c) When used in reference to radicals, the direction to “simplify” means to remove from the radicals any perfect roots that occur as factors. Figure 39 Now Work problem 131 Now Work problems 15 and 27 Two or more radicals can be combined, provided that they have the same index and the same radicand. Such radicals are called like radicals . Now Work problem 45 Simplifying Radicals (a) 32 162 16 2 42 = ⋅ = ⋅ = (b) 16 8 2 8 2 2 2 2 2 3 3 3 3 3 3 3 3 = ⋅ = ⋅ = ⋅ = (c) x x x x x x x x x x x 16 8 2 8 2 2 2 2 2 2 2 4 3 3 3 3 3 3 3 3 3 3 3 ( )( ) ( ) ( ) − = − ⋅ ⋅ ⋅ = − = − ⋅ = − ⋅ = − (d) x x x x x x x x x 16 81 2 3 2 3 2 3 2 3 5 4 4 4 4 4 4 4 4 4 4 4 ( ) ( ) = = ⋅ = ⋅ = EXAMPLE 3 ↑ Factor out 16, a perfect square. ↑ (2a) ↑ Factor out 8, a perfect cube. ↑ (2a) ↑ Factor perfect cubes inside radical. ↑ Group perfect cubes. Combining Like Radicals (a) 812 3 843 3 8 4 3 3 163 3 153 − + = − ⋅ + = − ⋅ + = − + =− (b) x x x x x x x x x x x x x x x x x 8 4 27 2 1 4 3 2 1 4 3 2 1 12 2 11 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ( ) ( ) + − + = + − ⋅ + = ⋅ + − ⋅ + ⋅ = − ⋅ + = + EXAMPLE 4
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