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An Introduction to Statistics
Activity for Lesson 4: Fractional ExponentsIn 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.
x1 = x
x2 = xx
x3 = xxx
x4 = xxxx
x5 = xxxxx
x-1 = 1/x
x-2 = 1/x2
x-3 = 1/x3
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.
x1/2x1/2 = x(1/2 + 1/2) = x1 = x
x1/3x1/3x1/3 = x(1/3 + 1/3 + 1/3) = x1 = x
x1/4x1/4x1/4x1/4 = x(1/4 + 1/4 + 1/4 + 1/4) = x1 = x
In other words:
x1/2 = sqrt(x)
x1/3 = cube root of x
x1/4 = fourth root of x
41/2 = 2
81/3 = 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 (xa/b)to be the bth root of x raised to the athpower. 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-1is "bound rather tightly" to the 6. To be on the safe side, parenthesesshould be used: 729^(6-1).
- 6 MATH 5 brings up 729.This then gives the 6th root of 729.