Use the order of operations to simplify the expression:
$4[2+6(6-4)]$
The expression to evaluate is \(4[2 + 6(6 - 4)]\). To simplify this, we follow the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule ensures that everyone arrives at the same result by specifying the sequence in which operations should be performed. First, we identify the innermost grouping symbols, which are the parentheses \((6 - 4)\), and proceed outward from there.
First, we calculate the expression inside the inner parentheses: \(6 - 4 = 2\). Replacing this result back into our expression, we get:
\[4[2 + 6(2)]\]
Next, we perform the operations inside the square brackets. According to the order of operations, multiplication comes before addition. Therefore, we calculate \(6 \times 2 = 12\). Our expression now becomes:
\[4[2 + 12]\]
Continuing inside the brackets, we perform the addition: \(2 + 12 = 14\). Now the expression simplifies to:
\[4 \times 14\]
Finally, we complete the multiplication: \(4 \times 14 = 56\).
Following the order of operations systematically, we have reduced the expression \(4[2 + 6(6 - 4)]\) to its final value of 56. Thus, \(\boxed{56}\).
The order of operations is essential in fields like computer programming and engineering, where expressions must be interpreted consistently by machines and humans alike. For instance, if an engineer is calculating the total load on a support beam, they might have an expression like \(10[5 + 2(10 - 8)]\), representing different components of the structural weight. By strictly following the order of operations, they ensure the beam is properly designed to safely carry the total load, preventing errors that could arise from calculating in the wrong order.