Problem 31 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ (10)(-6) $$

Step-by-Step Solution

Answer

Answer: -60

Step 1: Identify the given expression

The given expression is: $$ (10)(-6) $$ We need to find the value of this expression by performing the multiplication.

Step 2: Understand the multiplication of positive and negative numbers

Whenever we multiply a positive number with a negative number, the result is always a negative number. In this case, we need to multiply the positive number 10 with the negative number -6.

Step 3: Perform the multiplication

Now, multiply the two numbers and take the sign into account: $$ (10)(-6) = - (10)(6) $$ Since 10 times 6 is 60, the final result is: $$ - (10)(6) = -60 $$ The value of the given expression \((10)(-6)\) is -60.

Key Concepts

Understanding Integer Multiplication

When we encounter integer multiplication, we are essentially looking at the multiplication of whole numbers, which can be either positive or negative. This concept is the foundation for much of algebra and appears frequently in various mathematical contexts.

Let's explore the process step by step using an example from the exercise where we multiply a positive integer with a negative one. To multiply 10 (a positive integer) by -6 (a negative integer), we follow these guidelines:

• Recognize that multiplying an integer by another has the simple rule of 'multiplying their absolute values and assigning the sign'. The absolute value of a number is the value without considering its sign.
• For the product of a positive and a negative integer, we always end up with a negative result. This is because the product of different signs is negative.
• Compute the absolute values first: The absolute value of 10 is 10, and the absolute value of -6 is 6, so their product is 60.
• Assign the sign: Since we have one positive and one negative integer, our result will take the negative sign, yielding -60.

In our exercise, the final calculation looks like this: \[ (10)(-6) = -60 \] and demonstrates the rule that a positive multiplied by a negative will give a negative outcome.

The Role of Negative Numbers

Negative numbers are essentially numbers with a value less than zero, indicated by a minus sign \( - \). They are critical in representing various real-world scenarios like temperatures below freezing, debts, or decreases in stock market values.

Understanding how to work with negative numbers is crucial when dealing with integer multiplication. Here's why:

• Knowing the sign rules for multiplication can help you determine the sign of the product.
• Negative numbers can be thought of as 'direction' signs, where negative often means 'opposite'.
• In multiplication, if we multiply two negative numbers, their 'opposite' directions cancel out, and the result is a positive number.
• However, as with our example \( (10)(-6) \), a negative number multiplied by a positive number yields a negative result, because we are essentially taking the positive number in the opposite direction.

This understanding is fundamental, not just for multiplying single-digit integers, but for operations involving more complex algebraic expressions as well.

Multiplying Within Algebraic Expressions

When we extend the principle of multiplying integers to algebraic expressions, the rules remain consistent. An algebraic expression can contain variables, numbers, and operation symbols. Multiplication in algebraic expressions follows the same sign rules as integer multiplication.

In the exercise, \( (10)(-6) \) is a simple algebraic expression where the variables are absent. However, in more complex expressions, you might find variables mingled with integers. For instance, in an expression like \( x(-4) \) where \( x \) is a positive number, simply multiply the value of \( x \) by 4 and attach a negative sign to the result.

If you were to have two variables with assigned positive or negative values, such as \( x \) and \( y \), the multiplication rules based on their signs would apply the same way they do for integers. This concept is the cornerstone of simplifying expressions and solving equations in algebra.

In all cases, maintaining accuracy in the sign of the product is as important as the numerical value when you are multiplying within algebraic expressions. It can change the entire direction and meaning of an equation or inequality in algebra.