86 CHAPTER R Review of Basic Concepts Simplified Radicals An expression with radicals is simplified when all of the following conditions are satisfied. 1. The radicand has no factor raised to a power greater than or equal to the index. 2. The radicand has no fractions. 3. No denominator contains a radical. 4. Exponents in the radicand and the index of the radical have greatest common factor 1, unless this would require increasing the number of radicals in the expression. 5. All indicated operations have been performed (if possible). Simplified Radicals In working with numbers, we generally prefer to write a number in its simplest form. For example, 10 2 is written as 5, and - 9 6 is written as - 3 2 . Similarly, expressions with radicals can be written in their simplest forms. If the index of the radical and an exponent in the radicand have a common factor, we can simplify the radical by first writing it in exponential form. We simplify the rational exponent, and then write the result as a radical again, as shown in Example 7. (c) 23 81x5y7z6 = 23 27 # 3 # x3 # x2 # y6 # y # z6 Factor. = 23 27x3y6z613x2y Group all perfect cubes. = 3xy2z223 3x2y Remove all perfect cubes from the radical. S Now Try Exercises 71 and 83. EXAMPLE 6 Simplifying Radicals Simplify each radical. (a) 2175 (b) -325 32 (c) 23 81x5y7z6 SOLUTION (a) 2175 = 225 # 7 Factor. = 225 # 27 Product rule = 527 Square root (b) -325 32 = -325 25 Exponential form = -3 # 2 2n an = a if n is odd. = -6 Multiply. NOTE Converting to rational exponents shows why these rules work. 2 7 13 2 = 121/321/7 = 21/311/72 = 21/21 = 221 2 Example 5(f) 24 13 = 131/221/4 = 31/211/42 = 31/8 = 8 3 Example 5(g)
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