Fractional Exponents and Radicals

Fractional Exponents: Calculator Warning Calculate the following in your calculator: Not Exact. What is the exact value of 21/2? 2 ^ ( 1 ÷ 2 )

A Math Trick to Evaluate 2½ Exactly Evaluate the following using Exponential Laws: 21/2.21/2 When multiplying, if the bases are the same add the exponents 21/2 + 1/2 = 21 2 Now use this result to figure out what 21/2 equals.

What Does 2½ Exactly Equal? Since: The final answer makes sense because: 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: But the decimal will go on forever because it is an irrational number. For the exact answer just use:

More Fractional Exponents Change the following into a radical: 21/3 = Remember 21/2 is also called the square root of 2 The cube root of 2. The solution to x3 = 2

nth Roots The nth root of a number x is a number r which, when multiplied by itself n times equals x. Examples:

Radical v. Exponential Notation Rewrite the following roots as an exponential expression:

Square Root Notation Remember: The 2 is implied!

Roots in the Calculator Exponential Notation: 1 2 ^ ( 1 ÷ 6 ) Radical Notation: ( 6 ) ( 1 2 ) MATH is found in #5

Fractional Exponents with Numerators greater than 1 Approximate the following expression: 25/3 ≈ Find the exact value: Expand Rewrite in Radical Notation Simplify

Exponential v. Radical Notation bp/q = Generally b≥0

Example Evaluate the following without a calculator: Rewrite in Radical Notation Simplify By rewriting the fractional exponential expression into a radical expression, the arithmetic is more straightforward.