Evaluate (8/27)^(-1/3)(81/256)^(-1/4)

Problem

(8/27)(−1/3)*(81/256)(−1/4)

Solution

1. Apply the negative exponent rule to both terms by taking the reciprocal of the bases and changing the sign of the exponents.

(27/8)(1/3)*(256/81)(1/4)

2. Rewrite the bases as powers of prime numbers to simplify the fractional exponents.

(3/2)(1/3)*(4/3)(1/4)

3. Apply the power of a quotient rule by multiplying the exponents inside the parentheses by the outer exponent.

((3(3⋅1/3))/(2(3⋅1/3)))*((4(4⋅1/4))/(3(4⋅1/4)))

4. Simplify the exponents to reduce the expression to simple fractions.

(3/2)*(4/3)

5. Multiply the fractions and cancel the common factor of 3 in the numerator and denominator.

(3⋅4)/(2⋅3)

6. Divide the remaining terms to find the final numerical value.

4/2=2

Final Answer

(8/27)(−1/3)*(81/256)(−1/4)=2

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