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The law of exponents is a set of rules that describe how to handle mathematical operations involving exponents. When dividing like bases, we subtract the exponents, a principle formally expressed as \( a^{m} \div a^{n} = a^{m-n} \). This is crucial when simplifying algebraic expressions like \(\frac{9 z^{3}}{z^{2}}\), where both the numerator and denominator share the same base \(z\). According to the law, we subtract the exponent in the denominator from the exponent in the numerator, resulting in \(z^{3-2} = z^{1} = z\). Understanding this rule helps students break down complex expressions into something much more manageable.

Another aspect of the law of exponents is handling multiplication, \(a^{m} \times a^{n} = a^{m+n}\), and power to a power, \( (a^{m})^{n} = a^{m \times n} \), among others. These rules are the building blocks for simplifying algebraic expressions and mastering algebra.

The distributive property, sometimes known as distribution, is a property used to simplify mathematical expressions and equations. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then summing the products. Formally, it is expressed as \(a(b + c) = ab + ac\). This property extends to division as well; for instance, when you have a fraction such as \(\frac{6z}{3}\), it indicates that the division should be distributed to both the numerator and denominator, which simplifies to \(2z\) after dividing 6 by 3.

Applying the distributive property makes it possible to break down expressions with fractions and complex multiplication into simpler terms, facilitating further calculations or simplifications.

When simplifying algebraic expressions, combining like terms is a fundamental step towards the solution. Like terms are terms that have the same variable raised to the same power. For instance, in terms like \(9z\) and \(2z\), \(z\) is the common variable. To combine like terms, we simply add or subtract the coefficients (numbers in front of the variables) while keeping the variable part unchanged.

For example, if we have the terms \(9z\) and \(\-2z\), we add the coefficients 9 and \(-2\) together to get 7, and keep the variable \(z\), resulting in the single term \(7z\). This process is instrumental in helping us obtain the most simplified form of an expression. It's like tidying up a room by organizing scattered similar items into a single place!