- Simplify the innermost parentheses: \((13 - 7) = 6\).
- Simplify the contents of the square brackets: \([24 - 6] = 18\).
- Calculate the exponentiation: \(3^4 = 81\).
- Perform the final subtraction: \(81 - 18 = \boxed{63}\).
Explanation
- Analysis of the Expression
We are tasked with simplifying the mathematical expression \(3^4 - [24 - (13 - 7)]\). This expression involves exponentiation, subtraction, and multiple grouping symbols. To evaluate this correctly, we must follow the standard order of operations (often referred to by the acronym PEMDAS or BODMAS), which dictates that we first address expressions within parentheses and brackets, then proceed to exponents, and finally perform multiplication, division, addition, and subtraction in the appropriate order.
- Evaluating Innermost Parentheses
First, we address the innermost parentheses, which contain the expression \((13 - 7)\). Calculating the difference inside these parentheses, we get:
\[13 - 7 = 6.\]
Substituting this result back into our original expression, we have:
\[3^4 - [24 - 6].\]
- Evaluating Square Brackets
Next, we simplify the expression inside the square brackets \([24 - 6]\). Calculating this difference, we find:
\[24 - 6 = 18.\]
Now, our expression is simplified to:
\[3^4 - 18.\]
- Exponentiation and Final Calculation
Now we evaluate the exponentiation term \(3^4\), which means multiplying 3 by itself four times:
\[3^4 = 3 \times 3 \times 3 \times 3 = 81.\]
Finally, we perform the last subtraction operation:
\[81 - 18 = 63.\]
Thus, the simplified value of the original expression is \(\boxed{63}\).
Examples
Imagine you are managing a inventory of supplies. You start with a base amount of resources calculated as a power of 3, in this case, \(3^4 = 81\) units. From this total, you need to subtract a complex shipment requirement defined by a nested set of conditions: you must remove 24 units, but you also have a credit of 6 units from a previous return \((13 - 7)\). By using the order of operations, you calculate the net reduction \([24 - 6] = 18\) and correctly determine your remaining stock of 63 units. This logical approach helps in accurate resource management and ensures that no accounting steps are missed in complex calculations.