Evaluate.
$\left((-1)^2-3\right)^3-5 \cdot 4$
Answer (2)
First, evaluate the expression inside the parentheses: ( − 1 ) 2 − 3 = 1 − 3 = − 2 .
Then, raise the result to the power of 3: ( − 2 ) 3 = − 8 .
Next, perform the multiplication: 5 × 4 = 20 .
Finally, subtract: − 8 − 20 = − 28 . Therefore, the final answer is − 28 .
Explanation
Understanding the Expression We are asked to evaluate the expression ( ( − 1 ) 2 − 3 ) 3 − 5 ⋅ 4 . We will follow the order of operations (PEMDAS/BODMAS) to simplify this expression.
Simplifying Inside Parentheses First, we evaluate the expression inside the parentheses: ( − 1 ) 2 = 1 . Then, we subtract 3: 1 − 3 = − 2 . So, the expression inside the parentheses simplifies to − 2 .
Evaluating the Exponent Next, we raise − 2 to the power of 3: ( − 2 ) 3 = − 2 ⋅ − 2 ⋅ − 2 = − 8 .
Performing Multiplication Now, we evaluate the multiplication: 5 ⋅ 4 = 20 .
Performing Subtraction Finally, we subtract the result of the multiplication from the result of the exponentiation: − 8 − 20 = − 28 .
Final Answer Therefore, the value of the expression ( ( − 1 ) 2 − 3 ) 3 − 5 ⋅ 4 is − 28 .
Examples
Understanding how to evaluate expressions like this is fundamental in many areas, such as physics and engineering, where formulas often need to be calculated. For example, if you're calculating the potential energy of a system, you might have an equation that looks similar to this. Suppose the potential energy U is given by U = ( x 2 − 3 ) 3 − 5 × y , where x = − 1 and y = 4 . Plugging in these values, you would calculate U = (( − 1 ) 2 − 3 ) 3 − 5 × 4 = − 28 units of energy. This shows how order of operations is crucial in real-world applications.
To evaluate the expression ( ( − 1 ) 2 − 3 ) 3 − 5 ⋅ 4 , first, we simplify the expression inside the parentheses, yielding − 2 . Raising to the power of 3 gives − 8 , and after multiplying 5 ⋅ 4 to get 20 , we subtract to find the final result of − 28 .
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