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Chapter 5: Problem 58

For Problems \(45-64\), perform the indicated operations. $$ -7 n-[4 n-(6 n-1)] $$

Short Answer

Expert verified
The simplified expression is \(-5n - 1\).

Step by step solution

01

Distribute Negative Sign Outside Parentheses

The expression we need to evaluate is \(-7n - [4n - (6n - 1)]\).Begin by distributing the negative sign outside the square brackets to each term inside the brackets:\(-7n - [4n - 6n + 1]\) because \(- (6n - 1) = -6n + 1\).
02

Simplify Inside the Brackets

Now, simplify the expression inside the brackets:\(4n - 6n + 1\).This simplifies to:\(-2n + 1\).
03

Remove Brackets by Distribution Again

Substitute the simplified expression back into the main equation:\(-7n - (-2n + 1)\).Distribute the negative sign:\(-7n + 2n - 1\).
04

Combine Like Terms

Combine the like terms by adding the coefficients of \(n\):\(-7n + 2n = -5n\).Thus, the expression simplifies to:\(-5n - 1\).

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Key Concepts

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

Distributive Property
The distributive property is a fundamental algebraic concept that allows us to simplify complex expressions. It involves distributing one term across terms within parentheses. In this exercise, the expression starts with:
  • \(-7n - [4n - (6n - 1)]\)
With the inner parentheses, we must distribute a negative sign across the expression
  • \(4n - (6n - 1)\) which becomes \(4n - 6n + 1\).
This step utilizes the rule that multiplying by a negative sign flips the sign of each term inside the parenthesis.After simplifying the inside expression,
  • the task is to eliminate the brackets by correctly distributing through the entire bracketed term,
resulting in changing \(-[4n - 6n + 1]\) to \(-4n + 6n - 1\). This concept helps in systematically breaking down a complicated expression to make handling easier.
Simplifying Expressions
Simplifying expressions is the process of making algebraic expressions as compact and manageable as possible. It helps in reducing clutter and making calculations straightforward. In the given problem, once the distribution is complete, you should simplify the expression inside the brackets:
  • \(4n - 6n + 1\).
This boils down to adding or subtracting the coefficients of the same variable. In this exercise, subtract:
  • \(4n - 6n\), resulting in \(-2n\).
After simplification within the brackets, the expression becomes just another straightforward addition outside the square brackets. This step is crucial as it reduces the potential for errors in later calculations by making the expression more direct and easier to manage.
Combining Like Terms
Combining like terms is an essential process in algebra where we sum or subtract coefficients sharing the same variable. In this exercise, after appropriately distributing and simplifying, the expression evolves to:
  • \(-7n + 2n - 1\).
To combine the terms \(-7n\) and \(+2n\), simply add their coefficients:
  • \(-7 + 2\) which gives \(-5\).
Thus, the expression simplifies to \(-5n - 1\).This technique is crucial as it helps in consolidating an expression to its simplest form. It allows for cleaner solutions and is a vital skill in solving equations or simplifying any form of algebraic expression.

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Most popular questions from this chapter

For Problems \(113-120\), find the indicated products. Don't forget that \((x+2)^{3}\) means \((x+2)(x+2)(x+2)\). $$ (3 n-2)^{3} $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (3 a+2)(a+4) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (y+3)(y-7) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (x+4)(x-10) $$

A square piece of cardboard is 14 inches long on each side. From each corner a square piece \(x\) inches on a side is cut out as shown in Figure 5.18. The flaps are then turned up to form an open box. Find polynomials that represent the volume and the exterior surface area of the box.
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