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Chapter 4: Problem 8

Determine whether the terms are like or unlike. $$-2 a^{2} b, 6 a^{2} b$$

Short Answer

Expert verified
The terms \( -2 a^{2} b \) and \( 6 a^{2} b \) are like.

Step by step solution

01

Identify Variables and Powers in the First Term

In the term \( -2 a^{2} b \), the variable part is \( a^{2} b \). The variables are \( a \) and \( b \) with powers of 2 and 1 respectively.
02

Identify Variables and Powers in the Second Term

In the term \( 6 a^{2} b \), the variable part is also \( a^{2} b \). The variables are \( a \) and \( b \) with powers of 2 and 1 respectively.
03

Compare Variables and Powers

Upon comparing the variables and their powers in the two terms, we notice that they are the same. The variable parts in both terms are \( a^{2} b \), meaning the variables, their arrangement, and the powers are the same in both terms.

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Key Concepts

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

Variables and Powers
Understanding variables and powers is critical when comparing algebraic expressions. A variable is a symbol, often a letter like \( a \) or \( b \), that represents a number. Variables can take on various values depending on the situation.
In algebraic expressions, powers, also known as exponents, are used to show how many times a variable is multiplied by itself. For example, in the expression \( a^{2} \), the variable \( a \) is raised to the power of 2, meaning \( a \times a \).
Let's look at the terms \( -2 a^{2} b \) and \( 6 a^{2} b \):
  • In both terms, the variable \( a \) is raised to the power of 2, indicated by \( a^{2} \).
  • The variable \( b \) is raised to the power of 1, often simply written as \( b \).
Recognizing these variables and their respective powers allows us to effectively compare terms in algebraic expressions.
Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operational symbols. They act like sentences in language, expressing relationships between quantities.
The terms \( -2 a^{2} b \) and \( 6 a^{2} b \) are examples of algebraic expressions. Here, each expression consists of a coefficient, a number that multiplies the variable portion. In these examples, -2 and 6 are the coefficients.
When working with algebraic expressions:
  • It is important to identify the variables, coefficients, and their powers.
  • Look for common variables and understand how operations affect them.
    For instance, changing the coefficient does not change the variables' relationship.
This ability to decipher expressions helps in simplifying and solving complex mathematical problems.
Comparing Terms
The process of comparing terms is crucial in determining whether they are "like" or "unlike". Like terms have the same variable parts: the same variables raised to the same powers.
In the exercise provided, both terms, \( -2 a^{2} b \) and \( 6 a^{2} b \), share the variable part \( a^{2} b \):
  • The variables \( a \) and \( b \) appear with identical powers in both terms.
  • The arrangement of the variable part \( a^{2} b \) is consistent between terms.
Even though the coefficients, -2 and 6, are different, the terms are considered like terms because the variable parts are identical.
Recognizing like terms is especially useful when simplifying algebraic expressions or when combining terms to solve equations efficiently.

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