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Chapter 2: Problem 98

Combine like terms. [ 1.3] $$a b^{2}+2 a^{2} b-a b^{2}$$

Short Answer

Expert verified
2a^2b

Step by step solution

01

Identify Like Terms

Examine the given expression and identify the terms with similar variables and exponents. The expression is \[ a b^{2}+2 a^{2} b-a b^{2} \]. The like terms are: \[ a b^{2} - a b^{2} \].
02

Combine the Like Terms

Add or subtract the coefficients of the like terms identified in Step 1. For the like terms \[ a b^{2} - a b^{2} \] the result is 0. So after combining, they cancel out. Thus, the simplified expression is: \[ 2 a^{2} b \].

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Key Concepts

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

Algebraic Expressions
In algebra, we often deal with expressions that consist of numbers, variables, and operators like addition or subtraction. These are known as algebraic expressions. For example, in the expression \(ab^{2} + 2a^{2}b - ab^{2}\), we have variables \(a\) and \(b\) combined in various ways. Algebraic expressions form the foundation for many algebra problems, and understanding how to manipulate them is key. By breaking down an expression into its individual parts, you can better manage and solve it. Work through it step-by-step, and always remember that these expressions follow standard mathematical operations.
Like Terms
Like terms are terms in an algebraic expression that have the exact same variables raised to the same power. In the expression \(ab^{2} + 2a^{2}b - ab^{2}\), the terms \(ab^{2}\) and \(-ab^{2}\) are like terms because they both contain the same variables \(a\) and \(b^{2}\) raised to the same power. Understanding like terms is crucial for simplifying expressions. This step involves identifying and combining them. Once you spot like terms, you can add or subtract their coefficients. Here, \(ab^{2} - ab^{2} = 0\), which greatly simplifies the expression.
Simplifying Expressions
Simplifying expressions makes an algebraic expression easier to work with. This process usually involves combining like terms and reducing the expression to its simplest form. In the provided example, after identifying and cancelling the like terms \(ab^{2}\), the simplified form is \(2a^{2}b\). Simplifying expressions helps in solving equations or further manipulating the expressions in more complex problems. Always remember to carefully combine like terms step-by-step, and ensure no like terms are missed. The goal is to reach the most concise version of the expression, which can then be easily evaluated or further used.

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