(16/81)^-3/4 X

Jasonbradley

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Jun 16, 2024

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Simplifying (16/81)^(-3/4)

This expression involves fractional exponents, which can seem intimidating at first, but it's actually straightforward to simplify. Here's a breakdown of the steps:

Understanding Fractional Exponents

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

A fractional exponent like (m/n) represents taking the nth root and then raising it to the mth power.

Applying the Rules

Let's apply these concepts to simplify (16/81)^(-3/4):

1. Negative Exponent: A negative exponent means we take the reciprocal of the base raised to the positive version of the exponent. So: (16/81)^(-3/4) = (81/16)^(3/4)

2. Fractional Exponent: We can break down the fractional exponent: (81/16)^(3/4) = [(81/16)^(1/4)]^3

3. Finding the Root: We find the fourth root of both 81 and 16: [(81/16)^(1/4)]^3 = [(3/2)]^3

4. Final Calculation: Finally, we cube the result: [(3/2)]^3 = 27/8

Final Result

Therefore, (16/81)^(-3/4) simplifies to 27/8.