(8/27)^-4/3

Jasonbradley

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Jun 16, 2024

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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

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

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

3. Simplify the fraction: 1/[(2/3)^4] = 1/(16/81) = 81/16

Therefore, (8/27)^(-4/3) = 81/16