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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.

x1 = x
x2 = x•x
x3 = x•x•x
x4 = x•x•x•x
x5 = x•x•x•x•x

x-1 = 1/x
x-2 = 1/x2
x-3 = 1/x3

We can extend this to define what 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 roots such as a/b (xa/b) to be the bth root of x raised to the ath power. The extension of any real number to any real power goes even beyond numbers lesson 15.

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

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