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(27/64)^-2/3=

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

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

This problem involves working with fractional exponents. Here's how to simplify it:

Understanding Fractional Exponents

  • Negative exponent: A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. For example, x⁻² = 1/x².
  • Fractional exponent: A fractional exponent represents a combination of a root and a power. For example, x^(m/n) = (ⁿ√x)ᵐ, where:
    • n is the root (e.g., square root, cube root)
    • m is the power.

Step-by-Step Solution

  1. Apply the negative exponent rule: (27/64)^(-2/3) = 1 / (27/64)^(2/3)

  2. Apply the fractional exponent rule: 1 / (27/64)^(2/3) = 1 / (³√(27/64))²

  3. Simplify the cube root: 1 / (³√(27/64))² = 1 / (3/4)²

  4. Square the fraction: 1 / (3/4)² = 1 / (9/16)

  5. Divide by a fraction is the same as multiplying by its reciprocal: 1 / (9/16) = 1 * (16/9) = 16/9

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

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