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Chapter 2: Problem 49

Simplify each expression as completely as possible. $$6(m-2 n)+(6 m-2 n)+6 m(-2 n)$$

Short Answer

Expert verified
The simplified expression is: \(12m - 14n - 12mn\).

Step by step solution

01

Distribute terms

To simplify the expression, first distribute the terms in the parenthesis. Multiply each term inside the parenthesis by the factors outside:ewlineThe expression is: \(6(m - 2n) + (6m - 2n) + 6m(-2n)\)ewlineDistribute the \(6\) in the first term: \(6 \times m - 6 \times 2n = 6m - 12n\)ewlineDistribute the \(6m\) in the third term: \(6m \times -2n = -12mn\)
02

Write the expanded expression

After distribution, write the expanded expression:\(6m - 12n + 6m - 2n - 12mn\)
03

Combine like terms

Next, combine like terms. The terms \(6m\) and \(6m\) are similar, and \(-12n\) and \(-2n\) are also similar:\(6m + 6m - 12n - 2n - 12mnewlineCombine the similar terms: ewline 12m - 14n - 12mn\)

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Key Concepts

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

Distribution in Algebra
Distribution is an essential concept in algebra that helps you simplify expressions containing parentheses. It involves multiplying each term inside the parentheses by the term outside. Let's break it down with the provided exercise:
Consider the expression: \(6(m - 2n) + (6m - 2n) + 6m(-2n)\)
This might look complex, but distributing simplifies it. Apply distribution as follows:
  • For the first term, distribute the 6: \(6 \times m - 6 \times 2n = 6m - 12n\)
  • The second term is straightforward since there's no distribution needed: \(6m - 2n\)
  • For the third term, distribute 6m: \(6m \times -2n = -12mn\)
After distribution, the expression looks simpler and more manageable:
\(6m - 12n + 6m - 2n - 12mn\)
Combining Like Terms
After distribution, the next step in simplification is combining like terms. Like terms are those that have the same variables raised to the same power. Combining them involves adding or subtracting the coefficients while keeping the variables constant.
From the distributed expression: \(6m - 12n + 6m - 2n - 12mn\)
Identify the like terms:
  • \

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Most popular questions from this chapter

Simplify by combining like terms whenever possible. $$y^{2}-6 y+y^{2}-y$$

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