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When you encounter an expression that contains parentheses, the first step is to simplify the contents inside them. Parentheses are used in mathematics to group numbers or variables that should be considered as a single entity or operation. This operation should be performed first, according to the rules of arithmetic. For example, in the expression \((21 - 9)\), the subtraction operation inside the parentheses needs to be simplified before tackling the entire expression.
By simplifying just the parentheses, you change the initial expression to something more manageable. Here, subtracting 9 from 21 gives you 12, leading to a simpler version: \(78 - \frac{6(12)}{4}\). This sets the stage for further simplification of the entire expression.

Understanding the order of operations is crucial when simplifying expressions, especially when various mathematical operations are involved. The order in which operations are performed is known by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following this sequence ensures accurate results and avoids errors.
In our example, after simplifying the parentheses, multiplication comes next according to the order of operations. We multiply \(6\) by the number obtained from the parentheses, \(12\), resulting in \(72\). This tackled part of the expression accurately: \(78 - \frac{72}{4}\). Following the correct order allows you to systematically break down complex expressions into simpler parts.

Division in mathematical expressions might seem straightforward, but its placement in the order of operations is critical. It often comes intertwined with multiplication, sharing the same level of precedence. After handling any multiplication operations, division is carried out next. This keeps the calculations orderly and compliant with mathematical rules.
In our example, we reach the division step after multiplying, presented as \(\frac{72}{4}\). By dividing 72 by 4, you arrive at 18. The expression is then transformed into something even simpler: \(78 - 18\).
Recognizing division as an operation that balances with multiplication can help you accurately simplify expressions, ensuring that each step builds towards a correct and meaningful result.