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(27/64)^-2/3

2 min read Jun 16, 2024
(27/64)^-2/3

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

This expression involves both fractional exponents and negative exponents, which can seem confusing at first. Let's break it down step by step.

Fractional Exponents

A fractional exponent like (1/n) signifies a root. For example, x^(1/2) is the square root of x, and x^(1/3) is the cube root of x.

In our case, (-2/3) means we need to find both the cube root (1/3) and the square (2) of the base.

Negative Exponents

A negative exponent indicates reciprocal. This means we need to flip the base fraction. For instance, x^(-1) is the same as 1/x.

Applying the Concepts

  1. Flip the Fraction: Due to the negative exponent, we first find the reciprocal of (27/64), which is (64/27).

  2. Find the Cube Root: The cube root of 64 is 4, and the cube root of 27 is 3. So, (64/27)^(1/3) = (4/3).

  3. Square the Result: Finally, we square the result from the previous step, (4/3)², which equals (16/9).

Final Answer

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

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