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Understanding negative numbers is crucial for solving problems in mathematics, especially in integer multiplication. Negative numbers represent values less than zero, often depicted with a minus sign (-) before the number. When you multiply integers, the signs of the numbers determine the sign of the product.
A helpful rule to remember is:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

Thus, when you multiply two integers with different signs, the product is negative. Multiplying two integers with the same sign results in a positive product. By internalizing these rules, you can easily tackle integer multiplication problems, such as \(-2 \times 3\) in our example, which results in a negative product because the signs are different.

When faced with complex mathematical expressions, the order in which you carry out operations is important to solve the problem correctly. In mathematics, this is known as the order of operations, often remembered by the acronym PEMDAS:

• P – Parentheses first
• E – Exponents (i.e., powers and square roots, etc.)
• MD – Multiplication and Division (left-to-right)
• AS – Addition and Subtraction (left-to-right)

For the exercise \(-2(3)(-7)\), the expression includes only multiplication, so we can directly multiply from left to right. Following the order of operations ensures that you approach each component of the equation in a structured way, eliminating errors that might arise from prioritizing the wrong part of the expression.

Multiplying integers involves combining whole numbers, treating their signs as an integral part of the calculation. It's essential to multiply the numerical values and then evaluate the signs according to the rules of arithmetic.
Let's consider the original problem: \(-2 \times 3 \times (-7)\). Start by multiplying the first two integers, \(-2 \times 3\), giving \(-6\). This is because one number is negative, and the other is positive, which results in a negative product.
Next, take that result and multiply by the third integer: \(-6 \times (-7)\). Despite the larger absolute values, since both numbers have the same sign (negative), the product is positive, following our rule that a negative times a negative yields a positive. Hence, the final answer is \42\. This method highlights the interaction between numerical values and their signs, which is a cornerstone of understanding integer multiplication.