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Understanding the order of operations is crucial when working with mathematical expressions. It's the set of rules that tells us the sequence in which we should solve different parts of a math problem. The acronym PEMDAS is a common way to remember the hierarchy: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This hierarchy means that if an expression contains exponents, like the problem at hand with the expression \(z^{x}\), the exponentiation should be performed right after calculations inside parentheses, and before any other operations such as multiplication or addition.

Let's apply PEMDAS to the given problem \( (-2)^{2} \). There are no parentheses that involve operations to be carried out first, so we move to the next step which involves exponentiation. Remember, this rule is crucial because if you perform operations out of order, you could end up with a completely different answer, which would be incorrect.

Exponentiation is an operation involving two numbers; the base and the exponent. For example, in \(2^{3}\), 2 is the base and 3 is the exponent, indicating that 2 must be multiplied by itself twice more, for a total of three times, giving us 2*2*2=8.

When handling negative bases raised to an even exponent, like \( (-2)^{2} \), the negative sign is effectively 'cancelled out' because a negative number multiplied by itself an even number of times results in a positive number. Therefore, \( (-2)^{2} \) equals 4, since \( -2 \times -2 = 4 \). Conversely, if the exponent were odd, the result would maintain its negative sign, like \( (-2)^{3} = -8 \).

Moreover, exponentiation precedes multiplication and division in the order of operations, which is essential to remember when solving more complex expressions. This step is done after any parentheses have been dealt with but before calculating any products, quotients, sums, or differences that may be present.

Substituting variables is a fundamental part of algebra, allowing us to work with expressions in a more flexible and practical way. In the provided exercise, the expression \(z^{x}\) requires us to substitute the variables with the given numbers. So for \(x=2\) and \(z=-2\), we replace \(x\) and \(z\) accordingly, yielding \( (-2)^{2} \).

Accuracy in this step is key to obtaining the correct result. Make sure to replace each instance of the variable with the number it represents. Additionally, it is essential to keep in mind the signs of the numbers involved. If a negative sign isn't honored during substitution or any subsequent step, it can drastically alter the outcome of your calculation. The simplicity of this process lies in its methodical nature - once you accurately replace the variables, you can proceed to apply the appropriate operations as dictated by the order of operations.