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**An Introduction to Statistics**

**Activity for Lesson 4: Fractional Exponents**

In order to do some problems in today's assignment, an expanded definition of exponents needs to be developed. Recall from Numbers lesson 4 the definition and rules for exponentiation as follows. *x*^{1} = *x**x*^{2} = *x*•*x**x*^{3} = *x*•*x*•*x**x*^{4} = *x*•*x*•*x*•*x**x*^{5} = *x*•*x*•*x*•*x*•*x*

*x*^{-1} = 1/*x**x*^{-2} = 1/*x*^{2}*x*^{-3} = 1/*x*^{3}

We can extend this to definewhat *x* raised to a fractional exponent means as follows.

Using the fact that powers with common bases are multiplied, the exponents are added. Square roots were introduced in numbers lesson 10.

*x*^{1/2}*x*^{1/2} = *x*^{(1/2 + 1/2)} = *x*^{1} = *x**x*^{1/3}*x*^{1/3}*x*^{1/3} = *x*^{(1/3 + 1/3 + 1/3)} = *x*^{1} = *x**x*^{1/4}*x*^{1/4}*x*^{1/4}*x*^{1/4} = *x*^{(1/4 + 1/4 + 1/4 + 1/4)} = *x*^{1} = *x*

In other words:*x*^{1/2} = sqrt(*x*)*x*^{1/3} = cube root of *x**x*^{1/4} = fourth root of *x*

4^{1/2} = 2

8^{1/3} = 2

729 ^{1/6} = ((729)^{1/3})^{1/2} = (9)^{1/2} = 3

We can define a real number *x* raised to rational rootssuch as *a*/*b* (*x*^{a/b})to be the *b*^{th} root of *x* raised to the *a*^{th}power. The extension of any real number to any real power goes even beyondnumbers lesson 15.

Such roots can be calculated on the calculator three ways as follows.

- 729^6
^{-1}where the*x*^{-1}is used. This seems to bean exception to the general inability of the calculator to process exponentscorrectly. That can be explained by the fact that the*x*^{-1}is "bound rather tightly" to the 6. To be on the safe side, parenthesesshould be used: 729^(6^{-1}). - 729^(1/6)
- 6
**MATH****5**brings up 729.This then gives the 6th root of 729.

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- Copyright ©2003, Keith G. Calkins. Revised on or after October 2, 2003.