(64/27)^-2/3

Jasonbradley

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Jun 16, 2024

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

This problem involves simplifying an expression with a fractional exponent and a negative exponent. Here's how we can 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 type of root, and the numerator (-2) indicates the power. In this case, we have a cube root (3) and a square (2).

Applying the Rules of Exponents

1. Negative Exponent: The negative exponent indicates a reciprocal. So, (64/27)^(-2/3) is the same as 1/(64/27)^(2/3).

2. Reciprocal: Taking the reciprocal of a fraction simply flips the numerator and denominator. Therefore, 1/(64/27)^(2/3) becomes (27/64)^(2/3).

3. Fractional Exponent: Now, we apply the cube root and the square:

   • Cube Root: The cube root of 27 is 3 (3 * 3 * 3 = 27).
   • Cube Root: The cube root of 64 is 4 (4 * 4 * 4 = 64).
   • Square: Now we have (3/4)^2. Squaring both the numerator and the denominator, we get (9/16).

Final Solution

Therefore, (64/27)^(-2/3) simplifies to 9/16.