(27/64)^-2/3 In Fraction

Jasonbradley

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Jun 16, 2024

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Simplifying (27/64)^(-2/3)

This problem involves simplifying a fractional exponent. Let's break it down step by step.

Understanding Fractional Exponents

A fractional exponent like (-2/3) represents both a root and a power. The denominator (3) indicates the root to be taken (in this case, a cube root), and the numerator (2) indicates the power to which the result is raised.

Applying the Rules

1. Reciprocal: The negative exponent indicates a reciprocal. Therefore, (27/64)^(-2/3) is the same as (64/27)^(2/3).
2. Cube Root: We calculate the cube root of both the numerator and denominator:
   • ∛64 = 4
   • ∛27 = 3
3. Squaring: We square the results from the previous step:
   • 4² = 16
   • 3² = 9

Final Result

Putting it all together, we get:

(27/64)^(-2/3) = (64/27)^(2/3) = (∛64/∛27)² = (4/3)² = 16/9

Therefore, (27/64)^(-2/3) simplified as a fraction is 16/9.