(16/81)^-3/4

Jasonbradley

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Jun 16, 2024

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Simplifying (16/81)^-3/4

This article will guide you through simplifying the expression (16/81)^-3/4. We will utilize the properties of exponents to break down the problem into manageable steps.

Understanding the Properties of Exponents

Before we dive into the simplification, let's recall some key properties of exponents that we will use:

• Negative exponents: a^(-n) = 1/a^n
• Fractional exponents: a^(m/n) = (a^(1/n))^m = (n√a)^m
• Power of a fraction: (a/b)^n = a^n / b^n

Simplifying the Expression

1. Addressing the negative exponent:

   • Using the property a^(-n) = 1/a^n, we can rewrite the expression as:
   • (16/81)^-3/4 = 1 / (16/81)^(3/4)

2. Simplifying the fractional exponent:

   • Applying the property a^(m/n) = (n√a)^m, we get:
   • 1 / (16/81)^(3/4) = 1 / (⁴√(16/81))^3

3. Evaluating the roots:

   • ⁴√(16/81) = (⁴√16) / (⁴√81) = 2/3
   • Substituting this back into the expression: 1 / (2/3)^3

4. Final Simplification:

   • (2/3)^3 = 2^3 / 3^3 = 8/27
   • Therefore, 1 / (2/3)^3 = 1 / (8/27) = 27/8

Conclusion

By applying the properties of exponents, we successfully simplified (16/81)^-3/4 to 27/8. This process demonstrates how understanding exponent rules can help us navigate and solve complex expressions efficiently.