Evaluate the expression:
$(-9)-(-8)+2 \times 4^2$
To evaluate the expression \((-9) - (-8) + 2 \times 4^2\), we must follow the standard order of operations, often abbreviated as PEMDAS/BODMAS. This rule dictates the sequence of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Our expression consists of subtraction, addition, multiplication, and an exponent.
First, we address the exponentiation. The exponent in our expression is \(4^2\). Calculating this gives us:
\[4^2 = 4 \times 4 = 16\]
Now our expression simplifies to:
\[(-9) - (-8) + 2 \times 16\]
Next, we perform the multiplication. We have \(2 \times 16\):
\[2 \times 16 = 32\]
Our expression now becomes:
\[(-9) - (-8) + 32\]
Now we perform addition and subtraction from left to right.
First, handle the subtraction of the negative number: \(-(-8)\) is equivalent to adding 8:
\[-9 - (-8) = -9 + 8 = -1\]
Finally, add the result to the remaining term:
\[-1 + 32 = 31\]
Thus, the final value of the expression is \(\boxed{31}\).
Understanding the order of operations is vital in many real-life applications, such as programming and finance. For instance, if you are creating a budget sheet, you might have an initial balance and want to add or subtract various transactions based on specific rates or quantities. Suppose you have an initial balance of \(\$(-9)\) (perhaps an account overdraft) and you receive \(\$8\), then add a investment that grows at a rate of 2 times the square of a specific factor (let's say 4). Calculating \(\$(-9) - (-8) + 2 \times 4^2\) would accurately tell you your final balance is \(\$31\), allowing you to manage your finances correctly.