Find the product.
$4(-5)(-2)(-7)$
We are tasked with finding the product of four specific integers: \(4, -5, -2,\) and \(-7\). The expression to evaluate is \(4 \times (-5) \times (-2) \times (-7)\). The process involves multiplying these four numbers while keeping track of the signs according to the rules of multiplication for integers. A key rule is that the product of two negative numbers results in a positive number, while the product of a positive number and a negative number results in a negative number. Since there are three negative factors in this expression, and three is an odd number, we expect the final product to be negative.
To simplify the multiplication, we can group the factors in pairs. Let's group them as follows:
\[ [4 \times (-5)] \times [(-2) \times (-7)] \]
First, we multiply the first two factors:
\[ 4 \times (-5) = -20 \]
Next, we multiply the last two factors:
\[ (-2) \times (-7) = 14 \]
Now, we multiply these two intermediate products together:
\[ -20 \times 14 = -280 \]
Alternatively, we could multiply from left to right:
\[ 4 \times (-5) = -20 \]
\[ -20 \times (-2) = 40 \]
\[ 40 \times (-7) = -280 \]
Both methods confirm the same result.
Having carefully performed the multiplication, we have determined that the product of the given integers is \(-280\). The sign is negative as expected, which confirms our initial analysis. Thus, the final product is \(\boxed{-280}\).
In finance, tracking cash flows often involves multiplying factors like number of transactions, their average values, and potential changes in interest rates. For instance, if a company incurs multiple periodic fees or losses, they might represent these as negative values to track total balance changes over time. Understanding how to handle signs in arithmetic is a fundamental skill for maintaining accurate records and interpreting financial data in daily business operations.