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The order of operations is a fundamental concept in arithmetic expressions and ensures that everyone performs calculations the same way. It is often remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Following this order is crucial for getting the correct result.
Let's see how it works by breaking down the example expression \((6+2)^{2} + (-5+3)^{2}\).
First, you solve the operations inside the parentheses: calculate \(6+2\) to get 8, and \(-5+3\) to get -2. Next in line are the exponents, so you compute \(8^2\) and \((-2)^2\). Finally, perform the addition of the results from the exponentiation steps.

Exponents are a way to express repeated multiplication of the same number. For instance, \(n^2\) means \(n \times n\). They greatly simplify expressions, allowing complex multiplication to be written more compactly.
In our example, you have \(8^2\) and \((-2)^2\).
Calculating these involves:

• \(8^2 = 8 \times 8 = 64\)
• \((-2)^2 = (-2) \times (-2) = 4\)

A key point to remember is that the exponent applies only to the number it's immediately attached to. Parentheses play a crucial role here, indicating that the exponent applies to the entire expression inside them, keeping the calculations orderly and correct.
In our computations, note how squaring \(-2\) results in a positive value, since multiplying two negative numbers results in a positive product.

Negative numbers can often be tricky, but they are merely numbers with a "minus" sign indicating their position to the left of zero on the number line. Understanding how to work with them is vital for arithmetic expressions.
When combining negative numbers with operations like addition or multiplication, certain rules apply:

• Adding a negative number is equivalent to subtraction.
• Two negative numbers multiplied yield a positive result.
• Subtracting a negative is the same as adding a positive.

In the given exercise, you see \(-5 + 3\). Treat this as \(-5\) "plus" \(+3\), which is like moving rightward from \(-5\) on the number line, ending at \(-2\).
Even when squared, as in \((-2)^2\), the pair of negative signs multiply to make a positive value, ensuring the outcome aligns with our expected result in calculations including exponents.