Evaluating Fractional Bases Raised to Powers
Taking a fractional base like 1/3 and raising it to a power involves using the definition of exponents, being careful about negatives, and simplifying properly. Let’s evaluate 1/3 raised to the 4th power step-by-step.
Remember the Exponent Rules
Key exponent rules:
- ab = a x b
- a0 = 1
- a-n = 1 / an
These allow manipulation of exponential expressions.
Apply the Definition of Exponents
By the definition, a4 means to multiply a by itself 4 times:
a4 = a x a x a x a
So for our example:
(1/3)4 = (1/3) x (1/3) x (1/3) x (1/3)
Multiply the Fractions
Multiplying fractions involves multiplying the numerators and multiplying the denominators:
(1/3) x (1/3) x (1/3) x (1/3) = (1 x 1 x 1 x 1) / (3 x 3 x 3 x 3) = 1/81
Therefore, (1/3)4 = 1/81
Watch for Negative Exponents
Note when the exponent is negative, use the rule a-n = 1/ an and simplify:
(1/3)-4 = 1/(1/3)4 = 1/(1/81) = 81
Understanding exponent rules allows properly evaluating powers of fractional bases. With practice, this becomes straightforward. Correctly manipulating exponents is an important algebra skill applicable across many technical fields.
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