Question

Find the product. $$(-3)(-1)(4)(-6)$$

Short answer

The product of the given numbers is \(-72\).

Step-by-step solution

Multiply the first two numbers

First, determine the product of the first two numbers which are \(-3\) and \(-1\). This is done by simply multiplying the absolute values of the two numbers and applying the rule that the product of two negative numbers is a positive number. Therefore, the result of \((-3) \times (-1)\) is \(3\).

Multiply the result by the next number

Next, multiply the result by the third number which is \(4\). Now the operation will be \(3 \times 4\) which gives \(12\).

Multiply the product with the last number.

The final step is to multiply the result by \(-6\). Since we are multiplying a positive number by a negative number, our final result should be negative. Hence, \(12 \times -6 = -72\).

Key concepts

Negative Number Multiplication

Understanding how to multiply negative numbers is vital in grasping the broader topic of integer operations. When we multiply two negative numbers, such as in the problem \( (-3) \times (-1) \), we apply a fundamental rule: the product of two negatives is a positive. This might seem counterintuitive at first, but it's consistent with the idea of 'opposites'. Just as two wrongs don't make a right, two negatives do make a positive—well, mathematically speaking, at least!

To conceptualize this, imagine owing money as a negative amount. If you owe three debts of one dollar each (\( -3 \ times -1 \)), and all are canceled out (multiplied together), you essentially clear the slate, resulting in a positive outcome. Hence, \( (-3) \times (-1) = 3 \), illustrating the rule beautifully.

Multiplication of Integers

Multiplication of integers extends beyond just understanding negative numbers. Whether working with positives, negatives, or a combination of both, the rules are straightforward. With positive numbers, multiplication is as we learned initially: simply add the number to itself repeatedly according to the times it is being multiplied. However, when negatives enter the picture, the signs play a critical role.

When a negative and a positive integer are multiplied, the result is always negative. Think of it as a 'conflict of interest': the positive is attempting to increase value, the negative to decrease it, and tension results in a negative product. On the other hand, as seen in our example, the multiplication of two negatives results in a positive. These rules ensure consistency across all arithmetic operations and help solve complex equations, especially when dealing with variables that can hold various integer values.

Mathematical Operations

At the core of problem-solving in mathematics are the mathematical operations: addition, subtraction, multiplication, and division. Each operation has its own set of rules and applications, and mastering them opens up the vast world of mathematics. Particularly, multiplication is a fundamental operation that frequently pops up in various mathematical fields, including algebra, geometry, and calculus.

The steps to solve multiplication problems often involve more than one operation, gradually building towards the final answer. The ability to handle these operations efficiently is crucial for progress in mathematics education and beyond. It involves understanding properties such as the commutative property, which allows us to rearrange numbers when multiplying, and the distributive property, which can simplify multiplication across parentheses. By getting these operations down pat, students lay a sturdy foundation for tackling higher-level math problems with confidence.