Problem 5 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(-8\).

Step 2: 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.

Step 3: Write the Simplified Expression

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

Key Concepts

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’s 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.