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 3^{3},
to get the numerator and
we multiply 4×4×4, or 4^{3}, to get the denominator. Thus,
(3/4)^{3} = (3^{3})/(4^{3}).
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} = (5^{4})/(2^{4}) = 625/16
II. (- 3/4)^{2} = ((- 3)^{2})/(4^{2}) = 9/16
III. (1/(- 7))^{3} = (1^{3})/((- 7)^{3}) = 1/(- 343) = - 1/343