Problem 21 Add or subtract the monomials. ... [FREE SOLUTION]

Chapter 6: Problem 21

Add or subtract the monomials. $$2 a^{2}+b^{2}-6 a^{2}$$

Short Answer

Expert verified

-4a^2 + b^2

Step by step solution

Identify Like Terms

Look at the given expression: $2a^2 + b^2 - 6a^2$. Identify the like terms. Like terms are terms that have the same variable raised to the same power. Here, $2a^2$ and $-6a^2$ are like terms, while $b^2$ is a different term.

Combine Like Terms

Combine the like terms $2a^2$ and $-6a^2$. This gives: $2a^2 - 6a^2 = -4a^2$. The $b^2$ term remains unchanged as it does not have any like terms to combine with.

Write the Final Expression

Now, put together the combined terms and the remaining term: $-4a^2 + b^2$. Therefore, the simplified expression is $-4a^2 + b^2$.

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Key Concepts

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

Like Terms

To simplify expressions, the first step is to identify 'like terms'. Like terms are terms that have the same variables raised to the same power. For example, in the expression given, $2a^2 + b^2 - 6a^2$, the terms $2a^2$ and $-6a^2$ are like terms because both have the variable $a$ raised to the second power. In contrast, $b^2$ is not a like term with $a^2$ because it involves a different variable.

Combining Like Terms

Once you've identified the like terms in an expression, the next step is to combine them. For instance, in the given expression, combining $2a^2$ and $-6a^2$ results in$2a^2 - 6a^2 = -4a^2$. The finer detail is that you add or subtract the coefficients (the numbers in front of the variables) of the like terms. Since these are like terms due to having the same variable, their coefficients can be mathematically combined. The term $b^2$ remains as it is, because there are no other like terms to combine with it.

Monomials

A monomial is a type of algebraic expression that contains just one term. A term can be a constant, a variable, or a product of constants and variables raised to a power. In the expression $2a^2 + b^2 - 6a^2$, each of $2a^2$, $b^2$, and $-6a^2$ is a monomial. When combining like terms or simplifying expressions, recognize that each term stands alone initially but can interact with other like terms. However, we need to highlight that monomials do not need any addition or subtraction within; they're already simplified into their basic form.

Algebraic Expressions

An algebraic expression is a mix of numbers, variables, and operations without an equal sign. They're like mathematical phrases that represent values. In the context of the problem, we dealt with the algebraic expression $2a^2 + b^2 - 6a^2$. It's crucial to understand how these expressions work since you fundamentally decompose them into more manageable parts by identifying like terms and combining them. The ultimate goal is to simplify the expression to a form that's easier to understand or use in further calculations. After simplifying, the final expression $-4a^2 + b^2$ is more concise and understandable.

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91影视

Add or subtract the monomials. $$2 a^{2}+b^{2}-6 a^{2}$$

Short Answer

Step by step solution

Identify Like Terms

Combine Like Terms

Write the Final Expression

Key Concepts

Like Terms

Combining Like Terms

Monomials

Algebraic Expressions

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