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When simplifying expressions, it's important to follow the order of operations. This is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding and applying these rules ensures that you simplify expressions correctly and avoid errors.

Let's look at our example expression: \(\big[(-8)^{2} - 5^{2}\big] \div (-4 + 1)\). Notice how the brackets and operations inside must be resolved in a specific order.

The first step is to handle any exponents inside the brackets. After solving exponents, follow through with addition/subtraction or multiplication/division as the order dictates.

Simply put, always work from the inside out and strictly follow PEMDAS. Once all operations inside parentheses or brackets are done, move outward step-by-step.

Exponents represent repeated multiplication. For example, \((-8)^2\) means multiply -8 by itself: \((-8) \times (-8) = 64\). Similarly, \(5^2\) means multiplying 5 by itself: \((5 \times 5 = 25)\).

Understanding how to calculate exponents is crucial because they often appear in algebraic expressions.

In our example, we have the exponents \((-8)^2\) and \(5^2\). These need to be solved before moving on to other operations. Correct evaluation of these exponents gives us 64 and 25, respectively.

This step lays the foundation for the next steps, as it transforms our complex expression into a simpler one that can be tackled with basic arithmetic.

Basic arithmetic operations include addition, subtraction, multiplication, and division—the fundamental building blocks of math.

In our example, once the exponents were simplified to 64 and 25, we performed subtraction: \[ 64 - 25 = 39 \].

The next arithmetic step simplified the denominator: \[ -4 + 1 = -3 \]. Finally, the division of the numerator by the denominator wraps it up: \[ 39 ÷ (-3) = -13 \].

Each of these steps demonstrates how following the proper order of operations and breaking down complex equations into simpler arithmetic makes solving them more manageable.

Mastering these basics helps you solve more complicated math problems seamlessly.