\(\left(\frac{1}{3}-2\right)^3=\)
Answer (1)
Simplify the expression inside the parentheses: 3 1 − 2 = − 3 5 .
Cube the result: ( − 3 5 ) 3 = − 27 125 .
The final answer is − 27 125 .
Explanation
Understanding the Problem We are asked to evaluate the expression ( 3 1 − 2 ) 3 .
Simplifying the Parentheses First, we simplify the expression inside the parentheses: 3 1 − 2 = 3 1 − 3 6 = − 3 5 .
Cubing the Result Now, we cube the result: ( − 3 5 ) 3 = ( − 3 5 ) ⋅ ( − 3 5 ) ⋅ ( − 3 5 ) = − 27 125 .
Final Value Therefore, the final value is − 27 125 ≈ − 4.63 .
Examples
Understanding how to manipulate and evaluate expressions with fractions and exponents is fundamental in many areas, such as physics and engineering. For instance, when calculating the volume of a shrinking cube due to evaporation, you might encounter similar expressions. If a cube's side length is initially 3 1 meters and decreases by 2 meters, the new volume would be ( 3 1 − 2 ) 3 cubic meters. Evaluating this expression helps determine the final volume of the cube after the change.
