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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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