Problem 46 - Elementary and Intermediate Algebra

Question

Multiply. \(7 \cdot(-6) \cdot 5 \cdot(-4) \cdot 3 \cdot(-2) \cdot 1 \cdot 0\)

Step-by-Step Solution

Answer

0

Step 1: Identify the Numbers

List all the numbers to be multiplied: 7, -6, 5, -4, 3, -2, 1, 0.

Step 2: Understand the Property of Zero

Recall that any number multiplied by 0 is 0.

Step 3: Apply the Property of Zero

Multiply the sequence of numbers by 0. Since there is a 0 in the multiplication sequence, the entire product will be 0.

Key Concepts

Multiplication

Multiplication is a basic arithmetic operation that combines groups of equal sizes. In simpler terms, it is repeated addition. For example, multiplying 3 by 4 (written as 3 × 4) is the same as adding 3 four times (3 + 3 + 3 + 3). This yields the result of 12.

When you multiply positive and negative numbers, you need to remember some rules:
* A positive number times a positive number gives a positive result.
* A positive number times a negative number gives a negative result.
* A negative number times a negative number gives a positive result.

These rules make it clear that the sign of the product depends on the signs of the numbers you are multiplying. For instance, if you multiply 7 by -6, you first multiply the absolute values (7 × 6 = 42) and then apply the rule (positive × negative = negative), resulting in -42.

Sequences

In mathematics, a sequence is a list of numbers in a specific order. The numbers in a sequence are called terms. Sequences can follow various patterns or rules.

In the given exercise, the sequence of numbers to be multiplied is: 7, -6, 5, -4, 3, -2, 1, 0.

This is a finite sequence of eight terms. When dealing with sequences in multiplication, each term is multiplied by the next term in the sequence. For example:

• Multiply 7 by -6 (which equals -42)
• Then multiply -42 by 5
• Continue this process with the remaining terms

Sequences help structure problems and break them into smaller, manageable parts.

Zero Property in Multiplication

The Zero Property in multiplication states that any number multiplied by zero equals zero. This is a fundamental property in mathematics.

In the given exercise, you are asked to multiply multiple numbers, and one of those numbers is zero. According to the Zero Property:

• No matter how many numbers you multiply, as soon as zero is included, the final product will be zero.
• Think of it as dropping a stone into a pool with a hole - regardless of how big or how many stones you drop, once a stone hits the hole, everything flows through it.

This property simplifies the solution of many multiplication problems. In our example, although multiple numbers are being multiplied, one of them is zero. Therefore, the result is zero regardless of the other numbers.