Problem 56 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. Find the product of \(-7,-1,\) and \(0 .\)

Step-by-Step Solution

Answer

The product is 0.

Step 1: Identify the Expression

We are given the mathematical expression to find the product of three numbers: \(-7, -1,\) and \(0\). This means we need to multiply these numbers together.

Step 2: Apply the Multiplication Rule

According to the multiplication rule, any number multiplied by zero results in zero. Therefore, regardless of the other numbers in the product, if one of them is zero, the entire product will be zero.

Step 3: Calculate the Product

Multiply the numbers step-wise: first multiply \(-7\) and \(-1\), which gives \(7\) because multiplying two negative numbers results in a positive number. However, multiplying \(7\) by \(0\) gives \(0\).

Key Concepts

Multiplication Rule

When faced with a multiplication task, remember the basic principle known as the "Multiplication Rule." This rule is fundamental in arithmetic and states that when you multiply two numbers, you combine groups of one number by the quantity of the other number. But there's more to it than that.

• The Multiplication Rule simplifies calculations, especially when breaking them down into easier steps.
• Order is important: the rule works no matter which number comes first, thanks to the commutative property (e.g., \(a \times b = b \times a\)).
• This rule aids in understanding why multiplication is essentially repeated addition, where, for example, \(3 \times 4\) is the same as adding 3 four times: \(3 + 3 + 3 + 3\).

In essence, understanding this rule can unlock understanding of deeper mathematical concepts, and it demonstrates early number sense crucial for more complex equations.

Negative Numbers Multiplication

Multiplying negative numbers can often seem tricky, but it follows a logical pattern. To grasp it fully, consider this:

• When you multiply two numbers that are both negative, the product is positive: \(-a \times -b = a \times b\).
• This happens because the two negatives effectively cancel each other out, much like turning a negative into a positive notionally.
• However, when multiplying a positive number with a negative number, the result will always be negative: \(a \times -b = - (a \times b)\).

For students, these rules might seem counterintuitive at first, but with practice, the patterns become clearer. An example from our original problem is multiplying \(-7\) by \(-1\), which results in \(7\) as both negatives transform into a positive.

Zero Property of Multiplication

One of the simplest yet powerful properties in multiplication is the "Zero Property of Multiplication." This property is straightforward: when you multiply any number by zero, the result is always zero.

• This property applies universally, meaning it doesn't matter how large or small the number is; the result will always be zero when zero is a part of the multiplication.
• It is a crucial property that helps in quickly solving expressions that include zero as a factor.
• In our example exercise, no matter the result from multiplying \(-7\) and \(-1\), introducing \(0\) ensures the final product is \(0\).

This property simplifies calculations greatly, especially when dealing with larger expressions where it can immediately reduce complexity by eliminating terms due to the presence of zero.