What is the value of the expression when $n=3$ ?
$-2 n(5+n-8-3 n)$
To evaluate the expression \(-2n(5 + n - 8 - 3n)\) at \(n = 3\), we begin by substituting the value \(3\) for the variable \(n\) throughout the expression:
\[ -2(3)(5 + 3 - 8 - 3(3)) \]
This substitution allows us to focus on simplifying the numerical terms according to the order of operations.
Next, we simplify the terms inside the parentheses. Following the order of operations, we first perform the multiplication:
\[ 3(3) = 9 \]
The expression inside the parentheses becomes:
\[ 5 + 3 - 8 - 9 \]
Now, we calculate the sum and differences:
\[ 5 + 3 = 8 \]
\[ 8 - 8 = 0 \]
\[ 0 - 9 = -9 \]
So, the expression inside the parentheses simplifies to \(-9\).
Now that the parenthetical expression is simplified to \(-9\), we substitute this back into the original expression:
\[ -2(3) \times (-9) \]
First, multiply \(-2\) by \(3\):
\[ -2 \times 3 = -6 \]
Finally, multiply this result by \(-9\):
\[ -6 \times (-9) = 54 \]
The value of the expression when \(n=3\) is 54. Comparing this result with the given choices, the correct option is 54.
This calculation is similar to finding the total cost of an order. For instance, if you are buying 3 items that cost 5 dollars each, but you have a discount of 8 dollars plus a variable fee (3 times the number of items), you can use expressions like this to model your final bill. Understanding how to substitute and simplify these algebraic expressions is a foundational skill in finance, science, and engineering for predicting outcomes and managing resources.