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When simplifying expressions, understanding the concept of absolute value is crucial. The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction. For example, both \(3\) and \( -3\) have an absolute value of \(3\), because they are each three units away from zero.
In mathematical terms, for any real number \(x\), the absolute value is defined as:
\[|x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases}\]
In the given exercise, the absolute value is applied to the products \(2.5(-3)\) and \( -16\). The calculation \(2.5(-3)\) results in \( -7.5\), which is 7.5 units away from zero, thus its absolute value is \(7.5\). Likewise, \( -16\) is 16 units from zero, making its absolute value \(16\). Whenever you encounter an absolute value in an expression, you will handle it by evaluating it first before continuing with other arithmetic operations.

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks for simplifying mathematical expressions. It's important to execute these operations in the correct order, which is often guided by the rules of the order of operations, or PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In the context of the exercise, after evaluating the absolute values, you're left with the operation \(8.5 + 7.5 - 16\). Here, addition and subtraction are the operations to be performed. Remember, addition and subtraction are inverse operations; they undo each other. Adding \(8.5\) and \(7.5\) gives a sum of \(16\), from which subtracting \(16\) yields \(0\). This illustrates the canceling effect between the sum of certain numbers and their subsequent subtraction.

Exponents represent repeated multiplication. The exponent tells us how many times the base is multiplied by itself. For example, \(4^2\) means \(4 \times 4\), and is equal to \(16\).
It's important to note that the base can be positive or negative. If the exponent is an even number and the base is negative, the result will be a positive number because a negative number multiplied by itself an even number of times results in a positive product. For instance, \( (-4)^2 = (-4) \times (-4) = 16\).
In the exercise, \(4^{2}\) is evaluated within an absolute value. Since squaring a number always results in a non-negative number, the absolute value has no effect on the final result of \(4^{2}\). Thus, \(|-(4^{2})| = | - 16 |= 16\), demonstrating how exponents should be tackled before considering the absolute value operation in expressions.