Problem 24 - Introductory Algebra for College Students

Question

perform the indicated multiplication. $$(-6)(-3)(10)$$

Step-by-Step Solution

Answer

The result of the multiplication \(-6 \times -3 \times 10\) is \(180\).

Step 1: Identify the integers

Identify the integers that need to be multiplied which are \(-6\), \(-3\) and \(10\).

Step 2: Multiply the first two integers

The first two integers are \(-6\) and \(-3\). Multiply these two first. \(-6 \times -3\) gives \(18\) because the product of two negative integers is a positive integer.

Step 3: Multiply the result with the third integer

Now, multiply the result from the step 2 which is \(18\) with the third integer \(10\). So, \(18 \times 10\) equals \(180\).

Key Concepts

Negative Number Operations

Understanding the operations on negative numbers is crucial in basic algebra and other areas of mathematics. When we multiply negative numbers, it's important to remember a core rule: the product of two negative numbers is a positive number.

For example, when multiplying ewline(-6) by (-3), we interpret this as taking 6 instances of a negative 3, which in turn gives us a positive 18. This can be a bit counterintuitive, because you might expect that combining two 'negatives' would result in something even more negative. However, think of it as two 'wrongs' making a 'right' in the world of mathematics.

This rule can be a beacon when navigating through more complex algebraic expressions involving negative numbers.

Multiplying Integers

When multiplying integers, it is important to pay attention to their signs. The simple rules to remember are: if the signs are the same, the product is positive; if the signs are different, the product is negative.

Let's apply this to the multiplication of ewline(-6) and (-3). Both numbers are negative, so the product is positive, giving us 18. When we introduce a positive integer like 10 into the mix, following the same guidelines, the product of a positive number (ewline18) and another positive number (10) remains positive, resulting in 180.

It allows us to predict the sign of the product easily, which is incredibly important for solving more complex problems.

Keep the Signs in Mind

Always check the signs of the integers you're multiplying to predict the result's sign immediately.

Basic Algebra

In basic algebra, we often deal with unknown values and the relationships between these values. While our exercise only involved known integers, the principles apply to unknowns as well.

The order in which we approach multiplication does not affect the final product due to the associative property of multiplication. This means that ewline(-6)(-3)(10) can also be computed as ewline((-6)(-3))(10) or ewline(-6)((-3)(10)), and the result will always be the same. This property simplifies computation and offers flexibility in tackling algebraic expressions, especially when variables are involved.

Moreover, understanding that the multiplication of integers obeys certain rules and properties ensures a steady foundation as one progresses into more advanced topics in algebra.