Problem 62 - Fundamentals of Mathematics

Question

Perform each operation. $$ 5 \cdot(-70) $$

Step-by-Step Solution

Answer

The answer is -350.

Step 1: Understand the Operation

We have the operation \(5 \cdot (-70)\) which means we need to multiply 5 by -70. This is an example of multiplying a positive number by a negative number.

Step 2: Apply the Rule for Multiplying with Negatives

When multiplying a positive and a negative number, the result will be negative. So, the product of 5 and -70 will be negative.

Step 3: Calculate the Product

Multiply the absolute values of the numbers: \(5 \times 70 = 350\).

Step 4: Assign the Correct Sign

Since we are multiplying a positive number by a negative number, the product is negative. Therefore, \(5 \cdot (-70) = -350\).

Key Concepts

Multiplying Positive and Negative Integers

Multiplying integers involves applying specific rules especially when sign changes occur. When you multiply a positive integer by a negative integer, like in our exercise with 5 and -70, you always end up with a negative result. This is because the rule for multiplying numbers with opposite signs states: the product will be negative.

This rule is important to remember and is a fundamental aspect of integer arithmetic. To apply this rule, begin by multiplying the absolute values of the numbers involved, ignoring the signs initially. Finally, apply the sign rule where a positive multiplied by a negative yields a negative result. In this case, it means that multiplying 5 (positive) by -70 (negative) gives us -350.

Properties of Multiplication

Multiplication has several inherent properties that can simplify calculations and help inform the correct way to perform operations. These properties include the commutative property, associative property, identity property, and distributive property.

• Commutative Property: This states that the order in which you multiply numbers does not affect the product: if you multiply 5 by -70 or -70 by 5, the result, -350, remains the same.
• Associative Property: This suggests that how you group numbers when multiplying does not impact the product, which is more useful in complex calculations.
• Identity Property: Multiplying any number by 1 leaves the number unchanged, which helps verify the result if one factor is adjusted.
• Distributive Property: If you have a number outside a parenthesis, you can distribute it to each number inside, making operations such as \[(a(b + c)) = ab + ac\].

Understanding these properties helps in solving and checking mathematical problems, assuring the correct manipulation of integers in multiplication operations.

Integer Arithmetic

Integer arithmetic is the branch of arithmetic dealing with numbers that have no fraction or decimal part. In this form of calculation, signs play a crucial role, especially in operations like addition, subtraction, multiplication, and division.

Integers can be positive, negative, or zero. Familiarity with the sign rules is essential for achieving correct results. Here's a quick recap of integer arithmetic rules:

• Addition: Adding two positive integers or two negative integers results in a sum that carries the original sign. Adding numbers with different signs involves finding the difference between their absolute values.
• Subtraction: Subtracting involves adding the opposite. For instance, \(a - b\) becomes \(a + (-b)\).
• Multiplication and Division: Multiplying or dividing integers assumes selecting the right sign as explained in multiplying positive and negative numbers.

Mastering integer arithmetic is foundational for higher-level mathematics, providing the skills necessary for handling more complex mathematical models and problems.