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Chapter 5: Problem 5

For Problems \(1-24\), divide the monomials. $$ \frac{-16 n^{6}}{2 n^{2}} $$

Short Answer

Expert verified
The quotient is \(-8n^4\).

Step by step solution

01

Simplify the Coefficients

First, simplify the fraction of the numerical coefficients \(-16\) and \(2\). Divide \(-16\) by \(2\) which equals \(-8\). So, the fraction of coefficients becomes \(-8\).
02

Simplify the Variables

Next, simplify the variables by applying the quotient rule for exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, you have \(\frac{n^6}{n^2}\). Subtract the exponent of the denominator from the exponent of the numerator: \(6 - 2 = 4\). Thus, \(n^4\) is the result.
03

Write the Simplified Expression

Combine the simplified coefficient and variable from Steps 1 and 2. The resulting expression is \(-8n^4\).

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Key Concepts

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

Simplifying Coefficients
When dividing monomials, the first step is to simplify the coefficients, which are the numerical parts of the terms. In the example provided, we have the coefficients \(-16\) and \(2\) in the fraction \(\frac{-16 n^6}{2 n^2}\). Here, simplifying these coefficients involves basic division:
  • Divide \(-16\) by \(2\).
  • The solution to this is \(-8\).
This means that the numerical part of our monomial is now \(-8\). This simplification helps us reduce the complexity of the expression by focusing on factors that can be easily managed before dealing with variables.
Quotient Rule for Exponents
Once we have simplified the coefficients, the next step is dealing with the variables. In our exercise, we're working with \(n^6\) and \(n^2\). To simplify these, we apply the quotient rule for exponents. This rule is fundamental in algebra as it states:
  • When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
Applying this rule to our problem:
  • The exponent of \(n\) in the numerator is \(6\), and in the denominator, it's \(2\).
  • So, \(6 - 2 = 4\).
Therefore, the simplified expression for the variable part of the original monomial is \(n^4\). Using this rule efficiently helps simplify complex algebraic fractions.
Algebraic Expressions
In algebra, expressions that include numbers, variables, and operations are known as algebraic expressions. Dividing and simplifying such expressions, as we observed, involves both the numerical coefficients and the variable parts. Here鈥檚 why understanding them is crucial:
  • Coefficients provide the numerical value that controls the magnitude of the variable terms.
  • Variables, which we can often simplify using exponent rules, determine the expression's form.
In our problem, we started with \(-16n^6\) being divided by \(2n^2\). By separately simplifying the coefficients and applying the quotient rule to the exponents, we obtained the easier-to-work-with expression \(-8n^4\). This step-by-step simplification enables clearer understanding and makes solving more complex problems manageable.

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Most popular questions from this chapter

For Problems \(11-36\), find the indicated products by applying the distributive property and combining similar terms. Use the following format to show your work: $$ \begin{aligned} (x+3)(x+8) &=x(x)+x(8)+3(x)+3(8) \\ &=x^{2}+8 x+3 x+24 \\ &=x^{2}+11 x+24 \end{aligned} $$ $$ (5 a+2)\left(a^{2}+a-3\right) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (3-2 x)(9-x) $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (2 a+1)(a+6) $$

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ -4 x(2-3 x)(2+3 x) $$

For Problems \(113-120\), find the indicated products. Don't forget that \((x+2)^{3}\) means \((x+2)(x+2)(x+2)\). $$ (4 n-3)^{3} $$
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