Evaluate the expression:
$8 \div(-4) \times(-6)^2+7$
To evaluate the expression \(8 \div (-4) \times (-6)^2 + 7\), we must follow the standard order of operations, often referred to as PEMDAS/BODMAS. This rule dictates that calculations should be performed in the following sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Our expression contains an exponent, a division, a multiplication, and an addition.
First, we address the exponent. We have \((-6)^2\), which means multiplying \(-6\) by itself:
\[(-6)^2 = (-6) \times (-6) = 36.\]
Substituting this back into the original expression, we get:
\[8 \div (-4) \times 36 + 7.\]
Next, we perform the multiplication and division from left to right.
First, we divide \(8\) by \(-4\):
\[8 \div (-4) = -2.\]
Now, the expression is:
\[-2 \times 36 + 7.\]
Then, we perform the multiplication:
\[-2 \times 36 = -72.\]
The expression now simplifies to:
\[-72 + 7.\]
Finally, we perform the addition to find the result:
\[-72 + 7 = -65.\]
Thus, the final value of the expression is \(-65\).
Understanding the order of operations is crucial in many real-life applications, such as programming and engineering. For example, if you are writing a script to calculate the cost of a project where materials are priced differently depending on the volume (involving powers) and applying tax/discounts, the computer will follow these same rules. If you don't structure your equations correctly using parentheses, the computer might perform a multiplication before an exponent, leading to an incorrect total cost. Mastering this hierarchy ensures that your calculations remain accurate in every situation.