Problem 54 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ 8(5-12) $$

Step-by-Step Solution

Answer

Answer: $$-56$$.

Step 1: Perform the subtraction inside the parentheses

Calculate the value inside the parentheses: $$ 5-12 = -7 $$

Step 2: Multiply the result by 8

Now, multiply the result from step 1 by 8: $$ 8*(-7) = -56 $$ The value of the expression $$8(5-12)$$ is $$-56$$.

Key Concepts

Order of Operations

In algebra, the order of operations is a set of rules that reflect conventions about which procedures to perform first in a given mathematical expression. It is essential because it eliminates any ambiguity when interpreting expressions.

The order of operations can be remembered using the acronym PEMDAS:

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

In our scenario with the expression \(8(5-12)\), the parentheses indicate that the operation inside must be completed first. By following PEMDAS, we first tackle the subtraction inside the parentheses, which is \(5 - 12\). This step ensures we're proceeding correctly and laying the groundwork for multiplying the subsequent result.

Subtraction

Subtraction is a fundamental operation in mathematics, representing the process of removing objects from a collection. It's about finding the difference between numbers.

In the expression \(5 - 12\), we're subtracting a larger number, 12, from a smaller number, 5.
This results in a negative number, \(-7\).

Negative numbers come into play when you extend subtraction into the world of integers:

• If you subtract a larger number from a smaller one, the result is negative.
• Subtraction can thus describe changes that reduce the size of a value or describe positions on a number line that go below zero.

It's crucial to handle subtraction properly because it affects the subsequent operations if left uncalculated accurately.

Multiplication

Multiplication is another core arithmetic operation and is often symbolized by crossing (x) or dot (•) signs. Here, it is represented by placing a number adjacent to parentheses.

In our expression, the initial subtraction results in \(-7\), which must be multiplied by 8.
The rule when multiplying a positive number by a negative number is:

• The result is always negative.
• Multiply the absolute values of the numbers.

So, by calculating \(8 \times (-7)\), we find the product to be \(-56\).

This step solidifies our understanding of how a change in operation affects the result, reflecting the consistent effect of signs in multiplication between positive and negative numbers.