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The laws of exponents are rules that simplify expressions involving powers of the same base. These rules are essential in algebra, as they help in simplifying complex algebraic expressions efficiently.
When you are multiplying numbers or variables with the same base, as seen in the original exercise, the most applicable law is the multiplication law. This law states:

• When multiplying, you add the exponents of like bases: If you have \(a^m \times a^n\), it simplifies to \(a^{m+n}\).

In our example, the expression \((x^2 y)(y z)(x y z)\) involves multiplying same-base terms like \(x, y, \) and \(z\). Using the multiplication law, \(x^2 \times x\) simplifies to \(x^{2+1}\), which is \(x^3\). This law makes it far easier to manage expressions and reduce them to simpler forms.

Variable grouping is the process of reorganizing an expression to collect similar terms. This strategy is crucial for simplifying expressions, especially those involving multiple variables and operations.
In the given problem, the original expression was \((x^2 y)(y z)(x y z)\). By grouping like variables together, we can see each base and their respective exponents clearly. The expression is reorganized into \(x^2 \cdot x \cdot y \cdot y \cdot y \cdot z \cdot z\).
This helps us apply the laws of exponents correctly, as each group of similar variables can be easily combined. Grouping ensures that we don't miss any terms or make mistakes in applying algebraic operations. Additionally, it sets a clear visual structure for further simplification.

Algebraic expressions consist of variables, constants, and operations (add, subtract, multiply, divide). Simplifying such expressions helps to make them more manageable and easier to interpret. In our exercise, we start with a product of different algebraic expressions: \((x^2 y)(y z)(x y z)\).
The goal is to rewrite this product of variables and exponents into a more compact form.

• Compacting an expression involves utilizing the laws of exponents and strategic variable grouping.
• This reduces the complexity and helps clearly identify each component within the expression.

Simplifying algebraic expressions is not just an exercise in reducing terms; it's a fundamental skill in algebra that aids in solving more complex equations later on by providing clearer insight into the relationships between the variables involved.