Equivalent Expression Of (2mn)^4 / (6m^-3n^-2)

Hey guys! Today, we're diving into an exciting algebraic problem where we'll find an equivalent expression for the fraction (2mn)^4 / (6m^-3n-2). This involves simplifying expressions with exponents and fractions, which might seem tricky at first, but trust me, we'll break it down step by step so it's super clear. So, let's get started and unlock the secrets of exponents and fractions together! You'll be a pro in no time!

Breaking Down the Problem

To tackle this problem effectively, we need to understand the basic rules of exponents. When we raise a product to a power, each factor gets raised to that power. Also, when dividing terms with the same base, we subtract the exponents. Negative exponents indicate reciprocals. Keeping these rules in mind, let's dive into simplifying the expression.

Understanding the Key Concepts

Before we jump into the solution, let's quickly recap the key concepts we'll be using. Remember, when you have an expression like (ab)^n, it means a^n * b^n. This is crucial for dealing with the numerator in our problem. Also, when you divide terms with the same base, such as x^a / x^b, you subtract the exponents: x^(a-b). Don't forget about negative exponents! A term like x^-n is the same as 1 / x^n. These fundamental rules are the building blocks for simplifying algebraic expressions, and they'll be our best friends as we work through this problem. So, make sure you've got these concepts down, and let's get started!

Initial Expression

Our initial expression is:

$$\frac{(2 m n)^4}{6 m^{-3} n^{-2}}$$

This looks a bit complex, right? But don't worry, we'll simplify it step by step. Our goal is to get rid of the parentheses and negative exponents to make the expression cleaner and easier to understand. We'll use the rules of exponents that we just discussed to achieve this. The first thing we'll do is tackle the numerator, which has the term (2mn)^4. We need to distribute the exponent 4 to each factor inside the parentheses. This is where the rule (ab)^n = a^n * b^n comes into play. So, let's apply this rule and see what we get. It's like unlocking the first level of a game, and once we conquer this, the rest will follow more smoothly!

Step-by-Step Solution

Step 1: Simplify the Numerator

First, we simplify the numerator $(2 m n)^4$. Using the power of a product rule, we distribute the exponent 4 to each factor inside the parentheses:

$$(2 m n)^4 = 2^4 imes m^4 imes n^4 = 16 m^4 n^4$$

So, now we've transformed the numerator into a much simpler form: 16m⁴ⁿ4. We've successfully applied the power of a product rule, and this is a significant step forward. The expression looks less intimidating already, doesn't it? This is how we break down complex problems into manageable chunks. Remember, each step we take brings us closer to the final answer. Next, we'll incorporate this simplified numerator back into the original expression and see what to do with the denominator. Let's keep the momentum going!

Step 2: Rewrite the Expression

Now, we substitute the simplified numerator back into the original expression:

$$\frac{16 m^4 n^4}{6 m^{-3} n^{-2}}$$

Look how much cleaner it looks already! We've replaced (2mn)^4 with its simplified form, 16m⁴ⁿ4. This is a great example of how breaking down a problem into smaller parts can make it much easier to handle. Now, we have a fraction with terms involving m and n in both the numerator and the denominator. This sets us up perfectly to use the quotient of powers rule, which we discussed earlier. Remember, this rule helps us simplify expressions where we are dividing terms with the same base. In the next step, we'll focus on simplifying this fraction further by dealing with the exponents of m and n. We're making excellent progress – let's continue!

Step 3: Simplify the Coefficients

We can simplify the numerical coefficients first:

$$\frac{16}{6} = \frac{8}{3}$$

Simplifying fractions is a fundamental skill in mathematics, and it's super useful here. We've reduced 16/6 to its simplest form, which is 8/3. This makes our expression even cleaner and easier to work with. Remember, always look for opportunities to simplify numbers and fractions – it can save you a lot of headaches later on. Now that we've taken care of the coefficients, the next step is to focus on the variables, m and n. We'll use the rules of exponents to combine the terms in the numerator and the denominator. Let's keep moving forward and see how we can further simplify the expression!

Step 4: Simplify the Variables

Next, we simplify the variables using the quotient rule for exponents. For $m$, we have $m^4$ in the numerator and $m^{-3}$ in the denominator. Subtracting the exponents, we get:

$$m^{4 - (-3)} = m^{4 + 3} = m^7$$

And for $n$, we have $n^4$ in the numerator and $n^{-2}$ in the denominator. Subtracting the exponents, we get:

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

This is where the magic happens! We've successfully combined the m and n terms by subtracting the exponents. Notice how subtracting a negative exponent becomes addition – this is a common trick in algebra, so it's good to keep it in mind. We've transformed m^4 / m^-3 into m^7 and n^4 / n^-2 into n^6. Our expression is getting simpler and simpler with each step. Now, we're ready to put everything together and see what our final simplified expression looks like. Let's do it!

Step 5: Combine the Simplified Terms

Now, we combine the simplified coefficient and variable terms:

$$\frac{8}{3} m^7 n^6$$

So, there you have it! We've successfully simplified the original expression to (8/3)m⁷ⁿ6. This is a much cleaner and more manageable form. We started with a complex fraction with exponents and negative exponents, and through careful application of the rules of exponents and simplification, we arrived at our final answer. This process demonstrates the power of breaking down a problem into smaller, more manageable steps. Each step built upon the previous one, leading us to the solution. In the next section, we'll take a look at the final answer and discuss how it matches one of the given options. Let's wrap it up!

Final Answer

The equivalent expression is:

$$\frac{8 m^7 n^6}{3}$$

Matching the Options

Looking at the options provided, we can see that our simplified expression matches option A:

A. $\frac{8 m^7 n^6}{3}$

So, we've not only found the equivalent expression but also identified the correct option. Awesome! This confirms that our step-by-step simplification was accurate and effective. It's always a good feeling when your hard work pays off and you arrive at the right answer. Remember, the key to solving these types of problems is to take your time, apply the rules of exponents carefully, and break the problem down into smaller, more manageable steps. In the final section, we'll recap the entire process and highlight the key takeaways from this problem. Let's finish strong!

Conclusion

In this problem, we successfully simplified the expression $\frac{(2 m n)^4}{6 m^{-3} n^{-2}}$ to $\frac{8 m^7 n^6}{3}$. We achieved this by applying the rules of exponents step by step. First, we simplified the numerator by raising each factor inside the parentheses to the power of 4. Then, we rewrote the expression and simplified the numerical coefficients. Next, we used the quotient rule for exponents to simplify the variable terms. Finally, we combined all the simplified terms to get our final answer.

Key Takeaways

• Master the Rules of Exponents: Understanding the power of a product rule, quotient rule, and negative exponents is crucial for simplifying algebraic expressions.
• Break Down the Problem: Complex problems become easier when broken down into smaller, manageable steps.
• Simplify Step by Step: Simplify numerical coefficients and variable terms separately to avoid errors.
• Double-Check Your Work: Always double-check your calculations and simplifications to ensure accuracy.

So, guys, mastering these concepts and techniques will surely boost your confidence in tackling similar algebraic challenges. Keep practicing, and you'll become a math whiz in no time! Remember, every problem you solve is a step forward in your math journey. Keep up the great work!