Question

Find the product. $$-12 \cdot(-6)$$

Short answer

The product of \(-12\) and \(-6\) is \(72\).

Step-by-step solution

Identify the numbers

The numbers to be multiplied are -12 and -6.

Apply the rule of signs

When multiplying two negative numbers, the result is always positive. Therefore, it is the same as multiplying their absolute magnitudes. In this case, it's the same as doing \(12 \cdot 6\).

Perform the multiplication

Multiplication of 12 by 6 gives a product of 72.

Key concepts

Rule of Signs

Understanding the rule of signs is fundamental when dealing with the multiplication of integers, especially when negative numbers are involved. This rule is simple: when you multiply two numbers, and both have the same sign, the result will be positive. Conversely, if the two numbers have different signs, your answer will be negative.

For example, in the exercise \( -12 \cdot -6 \), both numbers are negative. Applying the rule of signs, we find the product is positive because the two signs are the same. Similarly, if we had a multiplication of a positive and a negative number, such as \( 12 \cdot -6 \), the answer would be negative.

This rule helps students avoid confusion and allows for quicker calculations once committed to memory, essentially turning a multiplication with negative numbers into a simpler operation.

Product of Integers

When we talk about the product of integers, we're referring to the result of multiplying these numbers together. Integers include all whole numbers and their negatives, plus zero. It's key to remember that multiplication is a quick way of adding the same number repeatedly.

Practical Example

If you were to multiply \(3 \cdot 4\), you're effectively adding 3 to itself 4 times: \(3+3+3+3\), which equals 12. In the context of our exercise, \( -12 \cdot -6 \), you're adding -12 to itself -6 times, which may seem confusing at first. However, by using the rule of signs, we simplify this to \(12 \cdot 6\), or adding 12 to itself 6 times, giving us 72. Having a strong grasp on how to work with integers, including negative ones, is crucial in algebra and beyond.

Multiplication of Absolute Magnitudes

The absolute magnitude of a number refers to its distance from zero on the number line, without considering its direction (positive or negative). Multiplying the absolute magnitudes of two negative numbers is an enlightening step towards obtaining the correct product.

In our given exercise, we disregard the negative signs of -12 and -6 temporarily and focus on their absolute magnitudes: 12 and 6. These are easier to work with because they're positive. After multiplying these absolute values together (\[ 12 \times 6 = 72 \]), we then apply the rule of signs to assign the correct sign to our answer. Since both original numbers were negative, our final answer to \( -12 \cdot -6 \) is a positive 72.

Training in the multiplication of absolute magnitudes helps students simplify complex problems and develop deeper mathematical understanding.