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Chapter 1: Problem 12

Simplify the given algebraic expressions. $$-x y^{2}-3 x^{2} y^{2}+2 x y^{2}$$

Short Answer

Expert verified
The simplified expression is \(xy^2 - 3x^2y^2\).

Step by step solution

01

Identify Like Terms

Look for terms in the expression that have the same variables raised to the same power. In this case, the like terms are \(-xy^2\) and \(2xy^2\).
02

Combine Like Terms

Add or subtract the coefficients of the like terms identified. Here, combining \(-xy^2\) and \(2xy^2\) gives \( (-1 + 2)xy^2 = xy^2\).
03

Rewrite the Expression

Combine the results of the previous step with any remaining terms to form the final expression: \(xy^2 - 3x^2y^2\).
04

Present the Simplified Expression

The simplified expression after combining like terms is \(xy^2 - 3x^2y^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
In algebra, identifying like terms is an essential skill in simplifying expressions. Like terms contain the same variables raised to the same powers. This makes them compatible for addition or subtraction, similar to how you would combine regular numbers. In the expression \(-xy^2 - 3x^2y^2 + 2xy^2\), the terms \(-xy^2\) and \(2xy^2\) are like terms because they share the identical variable combination \(xy^2\). Identifying such terms is like sorting similar items into groups before performing arithmetic operations.
Coefficients
Coefficients are the numerical parts of the terms in an algebraic expression. They tell you how many times the variables are being multiplied in the term. For example, in the term \(-3x^2y^2\), the coefficient is \(-3\). In \(-xy^2\), the coefficient is understood to be \(-1\) since any term with no visible number in front actually has 1 as a coefficient. Understanding coefficients is important when combining like terms because you'll need to sum them accordingly.
Variables
Variables are what make algebra dynamic and powerful. They are symbols, usually letters like \(x\) or \(y\), representing numbers that can take various values. In the expression \(-xy^2 - 3x^2y^2 + 2xy^2\), \(x\) and \(y\) are the variables. Variables can interact with each other in interesting ways such as being multiplied or raised to a power. This versatility enables the modeling of real-world problems using algebraic equations.
Simplification of Expressions
Simplification is the process of reducing an expression to its most concise and easily understandable form. This often involves combining like terms, which we've highlighted earlier. For instance, in the expression \(-xy^2 + 2xy^2 - 3x^2y^2\), first, identify and combine like terms \(-xy^2\) and \(2xy^2\) to get \(xy^2\). Next, you compile this result with any other unlike terms present, resulting in the simplified expression \(xy^2 - 3x^2y^2\). Simplification helps provide the clearest form of an expression, which is very useful for further mathematical operations or solving equations.

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Most popular questions from this chapter

Perform the indicated divisions. In the optical theory dealing with lasers, the following expression arises: \(\frac{8 A^{5}+4 A^{3} \mu^{2} E^{2}-A \mu^{4} E^{4}}{8 A^{4}},(\mu\) is the Greek letter mu.)

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