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When simplifying algebraic expressions, the concept of absolute value is essential and often encountered. The absolute value of a number is the distance of that number from zero on the number line, regardless of direction; it is always non-negative. For example, the absolute value of both 7 and -7 is 7, denoted as
\(|7| = 7\) and \(|-7| = 7\).
The overall role it plays in an expression is to remove any negativity from a number, making it 'absolute'. This becomes particularly important when combining with other operations, as it ensures clarity about the sign of the numbers involved. In essence, absolute value forms a sort of 'safety net', ensuring that you're working with the correct magnitude, especially when dealing with subtraction and addition in a larger expression.

Exponentiation is a mathematical operation, involving two numbers, the base and the exponent. The exponent denotes how many times the base is multiplied by itself. For instance, \(2^2\) means that 2 is being multiplied by itself, resulting in \(2 \times 2 = 4\).
A critical aspect to grasp is the behavior of exponents with negative bases. When a negative number is raised to an even exponent, the result is positive, e.g., \((-2)^2 = 4\), while an odd exponent would yield a negative result, e.g., \((-2)^3 = -8\). This is because multiplying a negative number by itself an even number of times results in a positive number. Understanding the rules of exponentiation is crucial for simplifying expressions as it can considerably change the signs within the algebraic expression, thereby affecting the solution.

Arithmetic operations are the cornerstones of simplifying algebraic expressions. Basic operations include addition, subtraction, multiplication, and division. When applying these to algebraic expressions, it's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the provided solution, once the absolute values and exponents were evaluated, the subsequent step involves performing subtraction and addition sequentially. For instance, starting with \(-7 - 4 + 4\), we first combine -7 and -4, because addition and subtraction are performed from left to right. This gives us \(-11\), then we add 4, resulting in \(-7\) as the final simplified result. Recognizing the proper sequence in which to perform arithmetic operations is fundamental in solving algebraic expressions accurately and efficiently.