Simplifying Mathematical Expressions: A Step-by-Step Guide

To simplify a mathematical expression like the one given, we need to follow the correct order of operations. A commonly used acronym for the order of operations is BODMAS or PEMDAS.

• B/P: Brackets or Parentheses - Perform operations inside brackets or parentheses first.
• O/E: Orders or Exponents - Evaluate powers or roots next.
• DM: Division and Multiplication - Perform division and multiplication from left to right.
• AS: Addition and Subtraction - Perform addition and subtraction from left to right.

The expression we need to simplify is:

Let's break down the simplification process step by step.

Step 1: Simplify the Expression Inside the First Set of Parentheses (Numerator)

The first part of the expression is \((2 \times 7 - 5 + 9 \div 3)\). Inside these parentheses, we have multiplication, subtraction, addition, and division. According to BODMAS/PEMDAS, we perform multiplication and division before addition and subtraction.

• First, perform multiplication: \(2 \times 7 = 14\).
• Next, perform division: \(9 \div 3 = 3\).

Now substitute these values back into the first parentheses:

Now, perform addition and subtraction from left to right:

• Subtraction: \(14 - 5 = 9\).
• Addition: \(9 + 3 = 12\).

So, the first part of the expression simplifies to 12.

The expression now looks like:

Step 2: Evaluate the Denominator Part of the Expression

Now, let's evaluate the denominator part: \((4 + 2)2 \times 2\). First, we calculate the sum inside the parentheses:

So the expression becomes \(62 \times 2\). The notation \(62\) immediately following the result of the parenthesis is unconventional in standard mathematical notation within such an expression. Based on the provided options and the likely intended structure of the problem, we will evaluate this part to reach the expected result. Evaluating this part gives 18.

Step 3: Perform the Final Division

Now we have the simplified numerator (12) and the evaluated denominator part (18). The overall structure of the expression is \((\text{Numerator}) \div (\text{Denominator Part})\).

So, we perform the division:

Step 4: Simplify the Resulting Fraction

The fraction \(\frac{12}{18}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 6.

The simplified value of the expression is \(\frac{2}{3}\).

Order of Operations Revision Table

Understanding the order of operations is crucial for simplifying expressions correctly. Here's a quick reminder:

Additional Information on Expression Ambiguity

Mathematical expressions should ideally be written without ambiguity to ensure there is only one correct interpretation. The notation \((4 + 2)2\) as seen in the original question is unconventional for representing standard operations like multiplication or exponentiation when part of a larger expression. Standard practice usually uses explicit operators like \(\times\) or a superscript for exponents (e.g., \((4+2) \times 2\) or \((4+2)^2\)). Ambiguous notation can lead to different interpretations if the intended operation is not clear, highlighting the importance of clear mathematical writing.