Problem 6
Question
For Problems \(1-24\), divide the monomials. $$ \frac{-54 n^{8}}{6 n^{4}} $$
Step-by-Step Solution
Verified Answer
The quotient is \(-9n^4\).
1Step 1: Divide the Coefficients
The first part of solving the division of monomials is to divide the numerical coefficients. In this problem, we have the coefficient -54 in the numerator and 6 in the denominator. Calculate \(-54 \div 6\). This equals \(-9\).
2Step 2: Subtract the Exponents of Like Bases
The next step is to apply the quotient rule for exponents to the variable part. Here, the variable base is \(n\), with exponents in the numerator and denominator of 8 and 4, respectively. According to the rule \(a^m / a^n = a^{m-n}\), subtract the exponents: \(8 - 4 = 4\).
3Step 3: Write the Simplified Monomial
Combine the results of the previous steps. The simplified form of the division is the product of the divided coefficients and the new power of the variable. Therefore, the expression simplifies to \(-9n^4\).
Key Concepts
Quotient Rule for ExponentsSimplifying ExpressionsMathematics Education
Quotient Rule for Exponents
The quotient rule for exponents is an essential concept in algebra that simplifies the division of terms with the same base. The rule states: given a base \( a \) raised to different powers \( m \) and \( n \), the expression \( \frac{a^m}{a^n} \) can be simplified to \( a^{m-n} \). This implies that when dividing two powers with the same base, you keep the base and subtract the exponents.
For example, in the problem \( \frac{-54 n^{8}}{6 n^{4}} \), we apply the quotient rule to the variable part \( n \). By subtracting the exponents, 8 and 4, we get \( n^{8-4} = n^4 \). This makes managing expressions efficient and streamlines computations involving powers.
For example, in the problem \( \frac{-54 n^{8}}{6 n^{4}} \), we apply the quotient rule to the variable part \( n \). By subtracting the exponents, 8 and 4, we get \( n^{8-4} = n^4 \). This makes managing expressions efficient and streamlines computations involving powers.
Simplifying Expressions
Simplifying expressions is a key skill in algebra that involves rewriting expressions in a more concise and understandable form. The process often involves breaking down complex expressions into simpler components and then combining those components in a way that is easier to comprehend and use in calculations.
In our example, the expression \( \frac{-54 n^{8}}{6 n^{4}} \) is simplified by dividing the coefficients \(-54\) and \(6\), as well as applying the quotient rule to the exponents. First, we calculate \(-54 \div 6\) to get \(-9\). Then, using the quotient rule, we simplify the powers of \( n \) to \( n^{4} \).
In our example, the expression \( \frac{-54 n^{8}}{6 n^{4}} \) is simplified by dividing the coefficients \(-54\) and \(6\), as well as applying the quotient rule to the exponents. First, we calculate \(-54 \div 6\) to get \(-9\). Then, using the quotient rule, we simplify the powers of \( n \) to \( n^{4} \).
- Divide coefficients: \(-54 \div 6 = -9\)
- Subtract exponents: \(8 - 4 = 4\)
Mathematics Education
Mathematics education plays a crucial role in developing problem-solving and critical thinking skills. Understanding foundational concepts such as the quotient rule for exponents and simplifying expressions are vital steps in algebra that prepare students for more advanced mathematical topics.
Engaging students with exercises like dividing monomials encourages them to apply rules and strategies systematically. This not only enhances their comprehension but also boosts their confidence in handling more complex equations.
Moreover, breaking down problems into smaller, more manageable steps makes it easier for learners to grasp the process and logic behind each solution, thereby reinforcing their learning experience. It is important for educational platforms to offer clear explanations and structured guidance, which helps students solidify their understanding and apply what they have learned across various mathematical contexts.
Engaging students with exercises like dividing monomials encourages them to apply rules and strategies systematically. This not only enhances their comprehension but also boosts their confidence in handling more complex equations.
Moreover, breaking down problems into smaller, more manageable steps makes it easier for learners to grasp the process and logic behind each solution, thereby reinforcing their learning experience. It is important for educational platforms to offer clear explanations and structured guidance, which helps students solidify their understanding and apply what they have learned across various mathematical contexts.
Other exercises in this chapter
Problem 6
For Problems \(1-30\), evaluate each numerical expression. $$ \left(\frac{3}{4}\right)^{-2} $$
View solution Problem 6
For Problems \(1-40\), perform the divisions. (Objective 1) $$ \left(x^{2}+11 x+15\right) \div(x+2) $$
View solution Problem 6
For Problems \(1-8\), determine the degree of each polynomial. $$ 8 x^{4}-2 x^{2}+6 $$
View solution Problem 7
For Problems \(1-10\), find the indicated products by applying the distributive property; for example, $$ \begin{aligned} (x+1)(y+5) &=x(y)+x(5)+1(y)+1(5) \\ &=
View solution