Simplifying Expressions: Find The Equivalent Form

by Alex Johnson 50 views

Have you ever encountered an expression that looks daunting at first glance but can be simplified into something much cleaner and manageable? In mathematics, we often deal with expressions involving exponents and fractions, and the key is to apply the rules of exponents systematically. Let's dive into how we can simplify a complex expression and find its equivalent form. This article will guide you through the step-by-step process of simplifying such expressions, making it easier to tackle similar problems in the future. So, whether you're a student brushing up on your algebra skills or just a math enthusiast, this guide will provide you with the tools and knowledge to simplify complex expressions with confidence.

Understanding the Problem

Before we jump into the solution, let's understand the expression we are dealing with. We have the expression (3m(-1)n2)^4 / (2m(-2)n)3, and our goal is to find an equivalent expression from the given options. This involves applying the rules of exponents, such as the power of a product rule, the power of a power rule, and the quotient of powers rule. These rules allow us to manipulate the exponents and simplify the expression step by step. By mastering these rules, you'll be able to tackle a wide range of algebraic expressions with ease. So, let's break down the problem and see how we can simplify this expression.

To start, we need to remember some fundamental rules of exponents:

  1. (Power of a Product Rule: (ab)^n = a^n b^n
  2. Power of a Power Rule: (am)n = a^(mn)
  3. Quotient of Powers Rule: a^m / a^n = a^(m-n)
  4. Negative Exponent Rule: a^(-n) = 1/a^n

These rules are the building blocks for simplifying expressions with exponents. Make sure you have a good grasp of these rules before proceeding. They will guide us through each step of the simplification process. Now, let's apply these rules to our expression and see how they work in practice.

Step-by-Step Simplification

Let's break down the simplification process step by step.

Step 1: Apply the Power of a Product Rule

The first step is to apply the power of a product rule to both the numerator and the denominator. This means we distribute the outer exponent to each factor inside the parentheses. For the numerator, we have (3m(-1)n2)^4, which becomes 3^4 * (m(-1))4 * (n2)4. For the denominator, (2m(-2)n)3 becomes 2^3 * (m(-2))3 * n^3. This step helps us to separate the terms and deal with them individually, making the simplification process more manageable. So, after applying the power of a product rule, we have:

Numerator: 3^4 * (m(-1))4 * (n2)4 Denominator: 2^3 * (m(-2))3 * n^3

Step 2: Apply the Power of a Power Rule

Next, we apply the power of a power rule, which states that (am)n = a^(mn). This means we multiply the exponents when a power is raised to another power. In the numerator, (m(-1))4 becomes m^(-1 * 4) = m^(-4), and (n2)4 becomes n^(2 * 4) = n^8. In the denominator, (m(-2))3 becomes m^(-2 * 3) = m^(-6). Now, let's rewrite our expression with these simplified exponents:

Numerator: 3^4 * m^(-4) * n^8 Denominator: 2^3 * m^(-6) * n^3

Step 3: Simplify Constants and Apply the Quotient of Powers Rule

Now, let's simplify the constants and apply the quotient of powers rule, which states that a^m / a^n = a^(m-n). We have 3^4 = 81 and 2^3 = 8, so our expression becomes (81 * m^(-4) * n^8) / (8 * m^(-6) * n^3). Now, we apply the quotient of powers rule to the variables:

  • For m, we have m^(-4) / m^(-6) = m^(-4 - (-6)) = m^(-4 + 6) = m^2.
  • For n, we have n^8 / n^3 = n^(8 - 3) = n^5.

Putting it all together, our simplified expression is (81 * m^2 * n^5) / 8. This step combines the constants and simplifies the variables using the quotient of powers rule, bringing us closer to the final simplified form. So, after this step, we have a much cleaner expression that is easier to understand and work with.

The Final Answer

After following these steps, we have simplified the expression (3m(-1)n2)^4 / (2m(-2)n)3 to (81m2n5)/8. Comparing this to the given options, we find that option B, (81 m^2 n^5)/8, is the correct equivalent expression. This final answer is the result of systematically applying the rules of exponents and simplifying each part of the expression. By breaking down the problem into manageable steps, we were able to arrive at the correct solution. So, the next time you encounter a complex expression, remember to apply these rules step by step, and you'll be able to simplify it with confidence.

Therefore, the correct answer is:

B. (81 m^2 n^5)/8

Conclusion

Simplifying expressions involving exponents can seem challenging, but by understanding and applying the fundamental rules of exponents, you can break down even the most complex expressions into simpler forms. We've seen how the power of a product rule, the power of a power rule, and the quotient of powers rule can be used systematically to simplify expressions. Remember, practice is key to mastering these concepts. The more you work with these rules, the more comfortable you'll become with applying them. So, keep practicing, and you'll be able to tackle any expression with confidence. If you're looking to further your understanding of exponents and algebraic expressions, there are many resources available online and in textbooks. Don't hesitate to explore these resources and continue learning. Math is a journey, and every step you take brings you closer to mastery.

For further learning on algebraic expressions, you can visit Khan Academy's Algebra section, a trusted resource for math education.