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Simplifying an algebraic expression means making it easier to read, or more "simplified". Think of it like cleaning up our room — we want everything in its place and neat! When we simplify, we look for parts of the expression that can be condensed or made smaller. In algebra, this can mean getting rid of unnecessary terms or putting things together so they make more sense.

For the given exercise, after distributing the fractions, the expression becomes cluttered with different terms and fractions. Simplifying the fractions (like turning \(\frac{3}{2} \cdot \frac{2}{3}\) into 1) is one way we tidy the expression. It makes numbers easier to work with and lays the groundwork for the next steps, such as combining like terms. Simplification usually involves straightforward arithmetic, so take your time and move step by step until everything looks nicer!

Combining like terms is like grouping similar items together. Imagine you're sorting your toys: you put all the robots in one box and all the dolls in another. In math, we do this by looking for terms that have the same variable(s) to the same power.

• For example, \(3x\) and \(5x\) are like terms because they both have the \(x\) variable.
• In contrast, \(3x\) and 4 are not like terms, as one has a variable and the other does not.

In the step-by-step exercise, we found like terms such as \(\frac{3}{2}x\) and \(\frac{1}{2}x\), which both have the variable \(x\). By adding these together, you simplify the expression and further organize it toward a cleaner form. Always remember: Only terms with the exact same variables and powers can be combined!

The distributive property is a handy tool that lets us break down problems and work on smaller parts before piecing it all back together. When you distribute, you multiply a single term by each term inside a set of parentheses. Think of it as sharing equally! For example, when you see something like \(a(b + c)\), you distribute \(a\) by multiplying it by both \(b\) and \(c\), giving you \(ab + ac\).

In the exercise, the distributive property is used to multiply each fraction by the terms inside their respective brackets. This means multiplying \(\frac{3}{2}\) by both \(x\) and \(\frac{2}{3}\) separately, and \(\frac{1}{2}\) by both \(x\) and 8 separately. This step is crucial for "breaking down" the expression so that the next steps, like simplifying and combining like terms, can be done easily!