Simplifying Quotients With Exponents A Step By Step Guide

In this article, we will delve into the solution of a specific algebraic problem, focusing on the simplification of a quotient involving exponents and variables. The question we aim to address is: What is the quotient $\frac{2 m^9 n^4}{-4 m^{-3} n^{-2}}$ in simplest form, assuming $m \neq 0$ and $n \neq 0$? This problem falls under the domain of algebraic expressions and requires a solid understanding of exponent rules and simplification techniques. We will break down the problem step-by-step, explaining the underlying principles and demonstrating how to arrive at the correct answer. This comprehensive guide will not only provide the solution but also enhance your understanding of similar algebraic manipulations.

Breaking Down the Problem

To effectively tackle this problem, we need to understand the fundamental rules of exponents. The key rules that apply here are:

1. Quotient of Powers Rule: When dividing terms with the same base, we subtract the exponents. Mathematically, this is represented as $\frac{a^m}{a^n} = a^{m-n}$.
2. Negative Exponent Rule: A term with a negative exponent can be rewritten as its reciprocal with a positive exponent. This is expressed as $a^{-n} = \frac{1}{a^n}$.

Applying these rules systematically is crucial to simplifying the given expression. We will first address the numerical coefficients and then handle the variables $m$ and $n$ separately.

Step-by-Step Solution

Let's start by rewriting the expression:

$$\frac{2 m^9 n^4}{-4 m^{-3} n^{-2}}$$

1. Simplify the Numerical Coefficients:

The numerical coefficients are 2 and -4. Dividing 2 by -4 gives us:

$$\frac{2}{-4} = -\frac{1}{2}$$

2. Simplify the Variable m:

We have $m^9$ in the numerator and $m^{-3}$ in the denominator. Applying the quotient of powers rule, we subtract the exponents:

$$m^{9 - (-3)} = m^{9 + 3} = m^{12}$$

3. Simplify the Variable n:

Similarly, we have $n^4$ in the numerator and $n^{-2}$ in the denominator. Applying the quotient of powers rule:

$$n^{4 - (-2)} = n^{4 + 2} = n^6$$

4. Combine the Simplified Terms:

Now, we combine the simplified numerical coefficient and the variable terms:

$$- \frac{1}{2} m^{12} n^6 = -\frac{m^{12} n^6}{2}$$

Therefore, the simplest form of the quotient is $-\frac{m^{12} n^6}{2}$.

Analyzing the Options

Given the options:

a. $-\frac{m^{12} n^6}{2}$ b. $-\frac{m^{27} n^8}{2}$ c. $6 m^{12} n^6$ d. $8 m^{12} n^6$

Our derived solution matches option a. The other options can be eliminated as they do not correctly apply the rules of exponents and simplification.

Common Mistakes to Avoid

When simplifying algebraic expressions involving exponents, several common mistakes can occur. Being aware of these pitfalls can help in avoiding them.

• Incorrectly Applying the Quotient of Powers Rule: A frequent error is adding the exponents instead of subtracting them when dividing terms with the same base. Remember, $\frac{a^m}{a^n} = a^{m-n}$, not $a^{m+n}$.
• Misunderstanding Negative Exponents: Negative exponents indicate reciprocals, not negative values. For example, $a^{-n} = \frac{1}{a^n}$, not $-a^n$.
• Arithmetic Errors with Coefficients: Simple arithmetic mistakes when dividing or multiplying coefficients can lead to incorrect answers. Always double-check the numerical calculations.
• Forgetting to Apply the Rule to All Variables: Ensure that the exponent rules are applied to each variable present in the expression.

Best Practices for Solving Algebraic Problems

To excel in solving algebraic problems, consider the following best practices:

• Understand the Fundamental Rules: A strong grasp of exponent rules, order of operations, and algebraic identities is essential.
• Break Down Complex Problems: Divide complex problems into smaller, manageable steps. This makes the problem less daunting and reduces the chance of errors.
• Show Your Work: Writing down each step of the solution process helps in tracking progress and identifying mistakes.
• Check Your Answer: After arriving at a solution, verify it by substituting values or using alternative methods.
• Practice Regularly: Consistent practice is key to mastering algebraic techniques. Solve a variety of problems to build confidence and skill.

Additional Examples and Practice Problems

To further solidify your understanding, let's explore a few additional examples and practice problems.

Example 1

Simplify: $\frac{15 x^5 y^{-2}}{3 x^2 y^3}$

Solution:

1. Simplify the coefficients: $\frac{15}{3} = 5$
2. Simplify x: $x^{5-2} = x^3$
3. Simplify y: $y^{-2-3} = y^{-5} = \frac{1}{y^5}$
4. Combine: $5 x^3 \frac{1}{y^5} = \frac{5 x^3}{y^5}$

Example 2

Simplify: $\frac{-8 a^{-4} b^6}{2 a^{-1} b^{-2}}$

Solution:

1. Simplify the coefficients: $\frac{-8}{2} = -4$
2. Simplify a: $a^{-4 - (-1)} = a^{-4 + 1} = a^{-3} = \frac{1}{a^3}$
3. Simplify b: $b^{6 - (-2)} = b^{6 + 2} = b^8$
4. Combine: $-4 \frac{1}{a^3} b^8 = -\frac{4 b^8}{a^3}$

Practice Problems

1. Simplify: $\frac{12 p^7 q^3}{-6 p^{-2} q^{-1}}$
2. Simplify: $\frac{21 m^{-3} n^5}{7 m^2 n^{-4}}$
3. Simplify: $\frac{-10 x^4 y^{-6}}{-2 x^{-1} y^2}$

Solving these problems will provide additional practice and help reinforce the concepts discussed.

Conclusion

In conclusion, simplifying algebraic expressions involving quotients and exponents requires a thorough understanding of exponent rules and careful application of these rules. By breaking down the problem into smaller steps, addressing coefficients and variables separately, and avoiding common mistakes, one can effectively simplify complex expressions. The solution to the given problem, $\frac{2 m^9 n^4}{-4 m^{-3} n^{-2}}$, is $-\frac{m^{12} n^6}{2}$, which corresponds to option a. Consistent practice and a solid grasp of fundamental principles are key to mastering algebraic simplification. Through this detailed explanation and additional examples, we aim to enhance your understanding and proficiency in this area of mathematics.

Keywords: simplifying algebraic expressions, quotient of powers rule, negative exponents, algebraic simplification, exponent rules, solving algebraic problems.