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(-1/27)^-2/3 Simplify

less than a minute read Jun 16, 2024
(-1/27)^-2/3 Simplify

Simplifying (-1/27)^(-2/3)

This expression involves both negative exponents and fractional exponents. Let's break down the simplification step-by-step.

Understanding the Rules

  • Negative Exponent: A negative exponent means taking the reciprocal of the base. For example, x⁻¹ = 1/x.
  • Fractional Exponent: A fractional exponent like (1/n) represents taking the nth root of the base. For example, x^(1/2) = √x. A fractional exponent like (m/n) combines these concepts: x^(m/n) = (√ⁿx)ᵐ.

Step-by-Step Simplification

  1. Deal with the Negative Exponent: (-1/27)^(-2/3) = 1 / (-1/27)^(2/3)

  2. Apply the Fractional Exponent: 1 / (-1/27)^(2/3) = 1 / (∛(-1/27))²

  3. Calculate the Cube Root: 1 / (∛(-1/27))² = 1 / (-1/3)²

  4. Square the Result: 1 / (-1/3)² = 1 / (1/9)

  5. Simplify the Division: 1 / (1/9) = 9

Therefore, (-1/27)^(-2/3) simplifies to 9.

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