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Powers of Fractions

The meaning of (3/4)3 is (3/4)×(3/4)×(3/4), or three-fourths of three-fourths of three-fourths. As shown in the SparkNote on Fractions, when we multiply fractions together, we multiply their numerators together and we multiply their denominators together. To evaluate (3/4)3 = (3/4)×(3/4)×(3/4), we multiply 3×3×3, or 33, to get the numerator and we multiply 4×4×4, or 43, to get the denominator. Thus, (3/4)3 = (33)/(43).

To take the power of a fraction, take the power of the numerator to get the numerator, and take the power of the denominator to get the denominator. To take the power of a mixed number, convert the mixed number into an improper fraction and then proceed as above.


Examples:

I. (5/2)4 = (54)/(24) = 625/16

II. (- 3/4)2 = ((- 3)2)/(42) = 9/16

III. (1/(- 7))3 = (13)/((- 7)3) = 1/(- 343) = - 1/343

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