87 R.8 Radical Expressions (b) 26 x12y3 = 1x12y321/6 = x12/6y3/6 = x2y1/2 = x22y (c) 29 163 = 29 63/2 = 163/221/9 = 61/6 = 26 6 S Now Try Exercises 89 and 91. EXAMPLE 7 Simplifying Radicals Simplify each radical. Assume all variables represent positive real numbers. (a) 26 32 (b) 26 x12y3 (c) 29 163 SOLUTION (a) 26 32 = 32/6 = 31/3 = 23 3 In Example 7(a), we simplified 26 32 as 23 3. However, to simplify A 26 x B 2 , the variable x must represent a nonnegative number. For example, consider the statement 1-822/6 = 31-821/642. This result is not a real number because 1-821/6 is not a real number. On the other hand, 1-821/3 = -2. Here, even though 2 6 = 1 3 , A 2 6 x B 2 ≠23 x. If a is nonnegative, then it is always true that am/n = a1mp2/1np2. Simplifying rational exponents on negative bases should be considered case by case. Operations with Radicals Radicals with the same radicand and the same index, such as 324 11pq and -724 11pq, are like radicals. On the other hand, examples of unlike radicals are as follows. 225 and 223 Radicands are different. 223 and 223 3 Indexes are different. We add or subtract like radicals using the distributive property. Only like radicals can be combined. EXAMPLE 8 Adding and Subtracting Radicals Add or subtract, as indicated. Assume all variables represent positive real numbers. (a) 324 11pq - 724 11pq (b) 298x3y + 3x232xy (c) 23 64m4n5 - 23 -27m10n14 SOLUTION (a) 324 11pq - 724 11pq = 13 - 7224 11pq Distributive property = -424 11pq Subtract.
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