Evaluate the expression:
$4 \times(-8)+6-(-2)^3$
To evaluate the expression \(4 \times (-8) + 6 - (-2)^3\), we must strictly adhere to the order of operations, often abbreviated as PEMDAS/BODMAS. This sequence dictates that we prioritize parentheses, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). Our expression consists of one exponentiation, one multiplication, one addition, and one subtraction.
First, we handle the exponentiation. The expression includes the term \((-2)^3\), which means \((-2)\) is raised to the power of 3. We calculate this as:
\[(-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8\]
Now, we substitute this back into our expression:
\[4 \times (-8) + 6 - (-8)\]
Next, we perform the multiplication. We multiply \(4\) by \(-8\):
\[4 \times (-8) = -32\]
Our expression now simplifies to:
\[-32 + 6 - (-8)\]
Finally, we perform the addition and subtraction from left to right.
First, add \(-32\) and \(6\):
\[-32 + 6 = -26\]
Then, subtract \(-8\) (which is equivalent to adding \(8\)):
\[-26 - (-8) = -26 + 8 = -18\]
Therefore, the final result is \(-18\).
Calculating with order of operations is essential in many real-world scenarios, such as managing a bank account or programming. For example, if you start with a balance of \(\$0\), spend \(\$8\) four times (\(4 \times -8\)), deposit \(\$6\), and then face a series of adjustments that result in a triple negative impact of \(\$2\) (\(-(-2)^3\)), understanding the precedence of these operations ensures your final balance is calculated correctly. In this case, your final balance would be \(\boxed{-\$18}\).