%5E-2%2F3.jpeg "(8/27)^-2/3")
Simplifying the Expression (8/27)^(-2/3)
This article aims to explain the process of simplifying the expression (8/27)^(-2/3). We will break down the steps and clarify the concepts involved.
Understanding Fractional Exponents
A fractional exponent like (-2/3) signifies both a root and a power. The denominator (3) represents the root, in this case, the cube root. The numerator (-2) represents the power. Therefore, (8/27)^(-2/3) can be rewritten as:
(8/27)^(-2/3) = (³√(8/27))^-2
Calculating the Cube Root
First, we need to calculate the cube root of (8/27).
- The cube root of 8 is 2 (2 * 2 * 2 = 8).
- The cube root of 27 is 3 (3 * 3 * 3 = 27).
Therefore:
(³√(8/27))^-2 = (2/3)^-2
Dealing with Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. So:
(2/3)^-2 = (3/2)^2
Final Calculation
Finally, we square the fraction (3/2):
(3/2)^2 = (3 * 3) / (2 * 2) = 9/4
Conclusion
Therefore, the simplified form of (8/27)^(-2/3) is 9/4.
This process demonstrates the application of fractional exponents, including their connection to roots and powers, as well as the handling of negative exponents.