(8/27)^(-2/3)

Jasonbradley

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Jun 16, 2024

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Understanding (8/27)^(-2/3)

This expression involves fractional exponents and negative exponents. Let's break it down step-by-step to understand how to calculate it.

Fractional Exponents

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

Negative Exponents

A negative exponent means taking the reciprocal of the base raised to the positive version of that exponent. For example, x^(-n) is the same as 1/x^n.

Applying the Rules

Now, let's apply these rules to (8/27)^(-2/3):

1. Negative exponent: (8/27)^(-2/3) = 1 / (8/27)^(2/3)

2. Fractional exponent: 1 / (8/27)^(2/3) = 1 / (cube root of (8/27))^2

3. Simplifying: The cube root of (8/27) is (2/3), so we have 1 / (2/3)^2

4. Squaring: 1 / (2/3)^2 = 1 / (4/9)

5. Dividing by a fraction is the same as multiplying by its reciprocal: 1 / (4/9) = 1 * (9/4) = 9/4

Conclusion

Therefore, (8/27)^(-2/3) is equal to 9/4.