Simplifying (8/27)^4/3
This expression involves both a fractional exponent and a negative exponent. Let's break down how to simplify it step by step.
Understanding the Exponents

Fractional Exponent: A fractional exponent like 4/3 represents both a power and a root. The numerator (4) indicates the power to which the base is raised, and the denominator (3) indicates the root to be taken. In this case, (8/27)^(4/3) means taking the cube root of (8/27) and then raising the result to the power of 4.

Negative Exponent: A negative exponent indicates reciprocal. For any base 'a' and exponent 'n', a^n is equal to 1/(a^n).
Simplifying the Expression

Deal with the negative exponent: (8/27)^(4/3) = 1/[(8/27)^(4/3)]

Calculate the power and root: 1/[(8/27)^(4/3)] = 1/[(∛(8/27))^4] = 1/[(2/3)^4]

Simplify the fraction: 1/[(2/3)^4] = 1/(16/81) = 81/16
Therefore, (8/27)^(4/3) = 81/16