Use the order of operations to evaluate this expression:
$7+(5-9)^2+3(16 \div 8)$
The value of the expression is □.
The objective is to evaluate the mathematical expression: \(7 + (5 - 9)^2 + 3(16 \div 8)\). To solve this correctly, we follow the order of operations, often referred to as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). We identify three distinct parts of the expression: a constant \(7\), a term with an exponent \((5 - 9)^2\), and a term with multiplication \(3(16 \div 8)\).
We first resolve the expressions inside the parentheses.
Substituting these back into the expression, we get: \(7 + (-4)^2 + 3(2)\).
Next, we address the exponent. The term \((-4)^2\) means \(-4 \times -4\). Since the product of two negative numbers is positive, we get \((-4)^2 = 16\). Now the expression becomes: \(7 + 16 + 3(2)\).
Now, we perform the multiplication. We have \(3(2)\), which is \(3 \times 2 = 6\). The expression now simplifies to: \(7 + 16 + 6\).
Finally, we complete the addition from left to right:
The final value of the expression is \(\boxed{29}\).
Order of operations is fundamental in fields like programming, engineering, and finance. For instance, if you are writing a script to calculate interest rates or material costs, the computer must execute calculations in the correct sequence to yield the right budget. Understanding these rules ensures that formulas are translated correctly into logic, preventing costly errors in real-world projects.