Problem 17 - Basic Engineering Mathematics

Question

Evaluate $$ 16 \div(2+6)+18[3+(4 \times 6)-21] $$

Step-by-Step Solution

Answer

The evaluated expression equals 110.

Step 1: Evaluate the Parentheses

First, work on the parentheses in the expression. Start with the innermost parentheses: 1. Calculate the addition in \(2 + 6\), which equals \(8\).2. Calculate the multiplication in \(4 \times 6\), which equals \(24\).3. Substitute the values back: \[16 \div 8 + 18[3 + 24 - 21]\]

Step 2: Simplify the Inner Parentheses

Now continue with simplifying inside the square brackets. Start inside the brackets:1. Perform the subtraction and the remaining addition: \(3 + 24 - 21 = 6\). 2. Substitute back: \[16 \div 8 + 18[6]\]

Step 3: Evaluate the Division

Divide \(16\) by \(8\): \[16 \div 8 = 2\]. Replace in expression: \[2 + 18[6]\].

Step 4: Evaluate the Multiplication

Multiply \(18\) by \(6\): \[18 \times 6 = 108\]. Replace in expression: \[2 + 108\].

Step 5: Perform the Final Addition

Add \(2\) to \(108\): \[2 + 108 = 110\].

Key Concepts

Understanding Parentheses

In mathematical expressions, parentheses play a crucial role. They tell us what operations to perform first, allowing complex problems to be tackled more easily. When you see parentheses, always start with the calculations inside. This is important because operations within parentheses need to be completed before addressing anything outside them.
This concept is critical in exercises like the one above. First, simplify the expression inside the parentheses. For example, in the expression given, begin by solving the simple addition within \(2 + 6\) which simplifies to \(8\). Next, handle other operations', like multiplication inside parentheses e.g., \(4 \times 6 = 24\).
Perform these steps accurately to prevent mistakes and ensure the final computation is correct. Thus, parentheses act as a guide, helping sequence the order of operations correctly.

Mastering Division

Once parentheses are handled, the next step according to the order of operations is division. Division is the process of determining how many times one number is contained within another and is denoted by the division symbol (\(\div\)).
For instance, once you solve \(16 \div 8\), you find that \(8\) fits into \(16\) exactly \(2\) times. The expression simplifies to \(2\) after performing the division. It's vital to execute division before moving onto other operations like addition and multiplication outside the parentheses, aligning with the universal rules of mathematical operations known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Exploring Multiplication

Multiplication often follows just after solving parentheses and division, but it shares the same level of priority as division in the order of operations. It’s a basic arithmetic operation that can significantly change the outcome of an expression when done correctly.
In our exercise, once we simplified the parentheses and division, the expression moved to multiplication \(18 \times 6\). This results in \(108\). By breaking down the process into separate and defined steps, you ensure that calculations are precise and accurate.

• This avoids errors and ensures results are reliable.
• It helps keep computations systematic and straightforward.

Multiplication is essential as it often modifies the values that are added or subtracted in subsequent steps.

Applying Addition

Addition is often one of the most straightforward operations in the order process but follows the more pressing operations of parentheses, division, and multiplication. This operation combines numbers and answers the question of how many units you have all together.
In the final stages of the problem above, once division and multiplication are complete, you add \(2\) to \(108\), resulting in a total of \(110\).
Addition brings together all previously calculated results, making it critical to the final solution. It is typically one of the last steps you undertake, as seen here, to obtain a comprehensive answer to a complex problem.