91Ó°ÊÓ

When simplifying algebraic expressions, one must first identify the like terms. But what are like terms? Like terms are components in an expression that have identical variables raised to the same powers. This means that the only differences between like terms should be their coefficients—numerical factors in front of the variables.

For instance, consider the expression \[ -xy^2 - 3x^2y^2 + 2xy^2 \]. Here, \(-xy^2\) and \(2xy^2\) are like terms because they both include the variables \(x\) and \(y\) raised to the powers 1 and 2, respectively. Their only difference lies in their coefficients, which are \(-1\) and \(2\).

By correctly identifying like terms, you pave the way for efficient simplification of algebraic expressions, as it allows you to focus only on those parts that are directly comparable.

Once you've sorted out which terms are alike, the next step is to combine them. This simplifies the expression and makes it more compact. Combining like terms essentially means summing their coefficients while keeping the variable part unchanged.

In our example, we had the expression:\[ -xy^2 - 3x^2y^2 + 2xy^2 \]. We identified \(-xy^2\) and \(2xy^2\) as like terms. So, we add their coefficients: \(-1 + 2 = 1\). Therefore, these two terms combine to become \(xy^2\).

The remaining term \(-3x^2y^2\) in the original expression doesn’t have a like term to combine with, so it stays as it is in the final simplified expression. Always remember to maintain any terms that do not have partners for combining, as removing them would alter the expression's value.

After combining like terms, you arrive at the stage of algebraic simplification. The main goal is to express the original expression in its most reduced form while maintaining the same value.

Through the process of identifying and combining like terms, we have simplified the initial expression \[ -xy^2 - 3x^2y^2 + 2xy^2 \] to become:\[ xy^2 - 3x^2y^2 \].

Simplifying an expression not only makes it easier to work with but also reveals clearer insights into the relationships between its components. This often results in more straightforward calculations in subsequent mathematical operations involving the expression. Remember that the simplified form is always relative to the terms that were present; terms that don't have duplicatable components remain unchanged, as was the case with \(-3x^2y^2\). This is why careful inspection of the expression at each step is crucial to maintaining its integrity.