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The first step in simplifying algebraic expressions involving fractions is to divide the coefficients. Coefficients are the numerical factors in terms of an algebraic expression. For example, in the expression \( \frac{-12 z^{3}}{3 z} \), -12 is the coefficient of the numerator and 3 is the coefficient of the denominator. To simplify, we just divide -12 by 3 to get -4. This process is just like basic arithmetic division, but it is crucial because it simplifies the numerical part of our expression, making the rest of the problem easier to tackle.

After simplifying coefficients, the next step is to divide the variables. Variables are the letters that represent unknown values. When the same variable appears in both the numerator and the denominator, we can simplify by 'dividing' these variables. This is done by subtracting the exponents, as per the laws of exponents for division. Take the expression \(z^{3}\) divided by \(z\) as an example. Since these variables are the same (both are 'z'), we subtract the exponent in the denominator from the exponent in the numerator: \(3-1 = 2\). Hence, \(z^{3} / z = z^{2}\). Understanding how to divide variables correctly is an essential skill in algebra, as it simplifies expressions to their most basic form.

Subtracting exponents is a part of variables division when working with algebraic expressions. Whenever we divide two like terms with exponents, we subtract the exponent of the denominator from the exponent of the numerator. If our expression was \(z^{3} / z\), we'd thus have \(z^{3-1} = z^{2}\). It's important to remember that this rule applies only when the base of the exponent (the variable 'z' in this case) is the same in both the numerator and the denominator. This step significantly reduces the complexity of expressions and is a key concept in algebra that can be applied to a variety of problems, including those with more complex expressions.