Problem 26 - Algebra 1

Question

Find the product. $$(5)(-2)(7)$$

Step-by-Step Solution

Answer

The product of \(5\), \(-2\) and \(7\) is \(-70\)

Step 1: Multiply the first two numbers

Start by multiplying the first two numbers. Multiply \(5\) by \(-2\) to get \(-10\).

Step 2: Multiply the result with the third number

Then, multiply the resulting \(-10\) from step 1 with the third number \(7\). You get \(-70\).

Key Concepts

Multiplication of Integers

Multiplication is one of the fundamental operations in mathematics. When dealing with integers, there are important rules to follow which make it easier. Integers are whole numbers that can be positive, negative, or zero.
When you multiply integers, you work with both the numbers and their signs.

• If you multiply two integers with the same sign, be it both positive or both negative, the result is always positive.
• If you multiply two integers with different signs, meaning one is positive and the other is negative, the result is negative.

This rule helps clarify why the multiplication of 5 and -2 in the original exercise results in -10. The first integer (5) is positive, and the second one (-2) is negative, so their multiplication will yield a negative result.

Order of Operations

When solving mathematical expressions, the order of operations is crucial. Without following this order, the same problem could have multiple solutions. The order is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Within the context of multiplication, like in our exercise \((5)(-2)(7)\), we start from left to right. This means you first multiply 5 by -2, and then multiply the result by 7. Skipping or rearranging steps can lead to incorrect calculations, so adhering to the order is essential.

Negative Numbers

Understanding negative numbers is key when multiplying integers. Negative numbers are less than zero and are indicated by a minus sign (-).
They have interesting properties during multiplication:

• When multiplying two negative numbers, the negatives cancel each other out, resulting in a positive product.
• However, when a negative number is multiplied with a positive number, the result is always negative as per the rule discussed previously.

In our example, after calculating 5 times -2 to get -10, we then have a negative product. Multiplying this negative result by a positive 7, we again are left with a negative product, resulting in -70. Understanding these properties helps prevent mistakes during calculations involving negative numbers.