Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$(-6)(-3)(10)$$

Step-by-Step Solution

Answer

The result of the multiplication of (-6), (-3) and (10) is 180.

Step 1: Understand the problem

The problem is asking to multiply three numbers together: -6, -3, and 10.

Step 2: Multiply the first two numbers

Multiply -6 and -3. The rule of multiplication states that the product of two negative numbers is a positive number. Therefore, the product of -6 and -3 is 18.

Step 3: Multiply the result with the third number

Now multiply 18 (obtained from the previous step) by 10. The product is 180.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, often found on the left side of zero on a number line. They appear in various real-world contexts like debts, temperatures below zero, or elevations below sea level. When dealing with negative numbers in mathematics, several rules apply, especially during arithmetic operations like addition, subtraction, and, most relevant here, multiplication.

Understanding these numbers is crucial because they behave differently from positive numbers. For example, adding a negative is like subtracting a positive. When multiplying, the sign of the numbers involved affects the outcome significantly.

Multiplication Rules

When it comes to multiplying numbers, specific rules help determine the sign of the product:

• If both numbers are positive, the product is positive.
• If both numbers are negative, the product is positive. This might be counterintuitive, but it helps to think of a negative times a negative as a double "reversal," bringing us back to positive.
• If one number is positive and the other is negative, the product is negative.

In our exercise, multiplying \(-6\times -3\) results in \(18\) because both numbers are negative.
This double reversal results in a positive number. Subsequently, multiplying this positive \(18\) by \(10\) remains positive.
Remember, multiplying by zero always yields zero, which is another important aspect of multiplication rules.

Arithmetic Operations

Arithmetic operations are basic mathematical calculations. They include addition, subtraction, multiplication, and division. Each has its rules and order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). Multiplication is a binary operation, meaning it involves two numbers to produce a product.

In the context of our exercise, arithmetic operations show us how to proceed step-by-step. First, perform the multiplication of two numbers, which involves the multiplication rules. Then use the product of the first operation to multiply with the third number. Following these steps carefully ensures no errors arise from neglecting the structure of operations.

Practicing these operations helps build a strong foundation for more complex mathematical concepts.