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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