Problem 3 - Fundamentals of Mathematics

Question

Determine the value of each of the following. $$(4+8) \cdot 2 $$

Step-by-Step Solution

Answer

The result is 24.

Step 1: Evaluate Inside the Parentheses

First, evaluate the expression inside the parentheses: \(4 + 8\).Calculate the sum: \(4 + 8 = 12\). The expression now is \(12 \cdot 2\).

Step 2: Multiply the Result

Now, multiply the result from Step 1 by 2:\(12 \cdot 2\).Perform the multiplication: \(12 \times 2 = 24\).

Key Concepts

Parentheses

When solving mathematical expressions, the order of operations is crucial for reaching the correct answer. Parentheses play a significant role in this process by determining which part of an expression should be calculated first.
Inside a mathematical problem, anything inside parentheses is essentially telling you to "do this part first". For example, in our exercise, within the expression \((4 + 8) \cdot 2\), the parentheses around \(4 + 8\) indicate that we should add these numbers before doing anything else.
Ignoring parentheses or resolving them incorrectly can lead to wrong answers. Always handle the content within parentheses before moving on to other operations, as they set the priority.

Addition

Addition is one of the basic arithmetic operations. It involves finding the total or sum by combining two numbers. In the context of our given expression, the addition operation happens inside the parentheses.
We have \(4 + 8\). Here, you simply find the sum of 4 and 8 which results in 12. This step simplifies the expression to \(12 \cdot 2\) after the addition is complete.
It's important to follow the sequence as determined by the parentheses to ensure that addition gives the correct intermediate result before proceeding to other operations like multiplication.

Multiplication

After dealing with parentheses and completing any addition or subtraction operations dictated by them, multiplication usually follows next according to the order of operations.
In our worked example, once we've determined that \(4 + 8 = 12\), we move to multiplication: \(12 \cdot 2\). Multiplication is simply repeated addition. So, mathematically speaking, this means adding 12 to itself two times, which results in 24.
Multiplication is often executed after parentheses and addition because of its position in the hierarchy of operations, helping us correctly solve expressions like \((4 + 8) \cdot 2\) efficiently and accurately.