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When working with mathematical expressions, the first step is often simplifying the parts within parentheses. Parentheses help indicate which operations should be performed first in an expression. They act as a guide, prioritizing calculations inside them over others.
For example, in the expression \( 8 \cdot[3+9 \cdot(5+2)] \), identify the innermost parentheses and solve the operation there first. The expression \((5+2)\) simplifies to 7.
Parentheses are crucial because they help clarify the order of operations and prevent ambiguity, ensuring everyone reads and solves the math problem the same way.

Once the operations inside the parentheses are complete, multiplication is performed next according to the order of operations rules. In the expression from the exercise, multiplication follows right after resolving the innermost set of parentheses.
After simplifying \((5+2)\) to 7 within \(8 \cdot[3+9 \cdot 7]\), the next task is to multiply the values found inside the brackets. Multiply 9 by 7, leading the expression to become \(8 \cdot [3+63]\).
It is essential to address multiplication before any addition, when both are present without parentheses, to maintain the correct results in calculations. Applying this order prevents possible mistakes and aligns with the standard set of mathematical rules.

After addressing multiplication within the brackets, the next step is to perform any addition present. Just as with multiplication, addition follows wherever it's indicated, but it must come after any multiplications and divisions in the sequence of operations.
In the step-by-step solution, after multiplying 9 by 7 to get 63, the expression became \(8 \cdot [3+63]\). The subsequent task required adding 3 and 63, which resulted in 66, thereby simplifying to \(8 \cdot 66\).
Understanding how addition fits into the order of operations promotes clarity and accurate results. By consistently following these steps, you maintain consistent, error-free calculations.