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Fractional Exponents and Radicals Fractional Exponents: Calculator Warning Calculate the following in your calculator: 1 2 2 1.41421 Not Exact. What is the 1/2? exact value of 2 2^(1÷2) ½ A Math Trick to Evaluate 2 Exactly Evaluate the following using Exponential Laws: 1/2. 1/2 2 2 = 1/2 + 1/2 2 When multiplying, if the bases are the same add the exponents 1 2 2 Now use this result to figure out what 21/2 equals. ½ What Does 2 Exactly Equal? Since: 2 2 2 2 12 2 2 12 12 2 2 12 12 2 The final answer makes sense because: 2 2 2 2 2 What is the number whose square is 2: x2 = 2 ? Irrational Numbers Radical numbers/Fractional Roots are typically irrational numbers (unless they simplify to an integer). Our calculator gives: 2 2 1/ 2 1.41421 But the decimal will go on forever because it is an irrational number. For the exact answer just use: 2 More Fractional Exponents Change the following into a radical: 1/3 2 Remember 21/2 is also called the square root of 2 = 3 2 The cube root of 2. 3 The solution to x = 2 nth Roots The nth root of a number x is a number r which, when multiplied by itself n times equals x. n x r if r x n Examples: 3 125 5 since 5 5 5 5 125 3 5 32 2 since 2 2 2 2 2 2 32 5 Radical v. Exponential Notation Rewrite the following roots as an exponential expression: Fourth root 4 Fifth root 5 Tenth root 10 Seventeenth root 17 nth root n x x 1 x x 1 x x 4 5 1 10 x x 1 17 x x 1 n Square Root Notation Remember: 2 x x The 2 is implied! Roots in the Calculator 16 12 12 6 Exponential Notation: 1 2 ^ ( 1 ÷ 6 ) ( 1 2 ) Radical Notation: ( x 6 ) x is found in MATH #5 Fractional Exponents with Numerators greater than 1 Approximate the following expression: 5/3 2 ≈ 3.174502 Find the exact value: 2 5 3 1 Expand 1 1 1 2 2 2 2 2 3 3 3 3 1 3 2 2 2 2 2 3 3 2 3 3 5 Simplify 3 3 Rewrite in Radical Notation Exponential v. Radical Notation b b q p/q p or = q b p Generally b≥0 Example Evaluate the following without a calculator: Rewrite in Radical Notation 5 6 64 6 Simplify 2 5 64 5 32 By rewriting the fractional exponential expression into a radical expression, the arithmetic is more straightforward.