Equivalent Expressions For (4mn/(m^-2 N^6))^-2

Hey guys! Let's break down this math problem together. We've got the expression $(\frac{4 m n}{m^{-2} n^6})^{-2}$, and we need to find which of the given options is equivalent. Don't worry, it might look a bit intimidating at first, but we'll take it step by step and make it super clear. Math can be fun, I promise!

Initial Expression: $(\frac{4 m n}{m^{-2} n^6})^{-2}$

Okay, so let's start by writing down our initial expression. This is key so we know what we're working with. The expression we're dealing with is: $(\frac{4 m n}{m^{-2} n^6})^{-2}$. We have a fraction raised to the power of -2. Remember that negative exponents mean we're dealing with reciprocals, and exponents outside parentheses affect everything inside. This is super important to keep in mind as we move forward. We'll use some exponent rules and simplify this expression bit by bit. Think of it like untangling a knot – slow and steady wins the race!

Step-by-Step Simplification

First things first, let's tackle that negative exponent outside the parentheses. A negative exponent means we take the reciprocal of the fraction and change the sign of the exponent. So, $(\frac{4 m n}{m^{-2} n^6})^{-2}$ becomes $(\frac{m^{-2} n^6}{4 m n})^{2}$. See? We just flipped the fraction, and now the exponent is positive. This is a classic move in simplifying these kinds of problems.

Next, let’s simplify inside the parentheses before we square anything. We have terms with 'm' and 'n' in both the numerator and the denominator. Remember the rule: when dividing terms with the same base, you subtract the exponents. So, for the 'm' terms, we have $m^{-2}$ in the numerator and $m^1$ (which is just 'm') in the denominator. Subtracting the exponents, we get -2 - 1 = -3. So, we have $m^{-3}$. For the 'n' terms, we have $n^6$ in the numerator and $n^1$ (which is just 'n') in the denominator. Subtracting the exponents, we get 6 - 1 = 5. So, we have $n^5$. Putting it all together, our fraction inside the parentheses now looks like $\frac{m^{-3} n^5}{4}$.

Now our expression is $(\frac{m^{-3} n^5}{4})^{2}$. We're getting there! The next step is to apply the exponent of 2 to everything inside the parentheses. This means we raise each term to the power of 2. So, $m^{-3}$ becomes $(m^{-3})^2$, which is $m^{-3*2} = m^{-6}$. Then, $n^5$ becomes $(n^5)^2$, which is $n^{5*2} = n^{10}$. And finally, 4 becomes $4^2 = 16$. So, our expression now looks like $\frac{m^{-6} n^{10}}{16}$.

But wait, we're not quite done yet! We have a negative exponent on the 'm' term. Remember that $m^{-6}$ is the same as $\frac{1}{m^6}$. So, we can rewrite our expression as $\frac{1}{m^6} * \frac{n^{10}}{16}$, which simplifies to $\frac{n^{10}}{16 m^6}$. And there we have it! We've simplified the original expression step by step, and now we have our final form.

Final Simplified Form: $\frac{n^{10}}{16 m^6}$

So, after all that simplifying, we've arrived at the expression $\frac{n^{10}}{16 m^6}$. This is our final answer. Now, let's take a look at the options provided and see which one matches.

Matching the Simplified Expression with the Options

Alright, so we've simplified our original beast of an expression to the much friendlier form of $\frac{n^{10}}{16 m^6}$. The goal now is to see which of the answer choices given to us actually matches this. This is like the moment in a puzzle where you find the piece that fits perfectly. Let's go through the options one by one and compare them to our simplified expression. This is where attention to detail really pays off!

• Option A: $\frac{n^6}{16 m^8}$

  Okay, let's look closely at Option A. It's $\frac{n^6}{16 m^8}$. Notice that the exponent on the 'n' in the numerator is 6, and the exponent on the 'm' in the denominator is 8. But in our simplified expression, we have $n^{10}$ in the numerator and $m^6$ in the denominator. So, the exponents don't match up. This means Option A is not equivalent to our simplified expression. We can cross this one off the list! It’s all about matching those exponents, guys.

• Option B: $\frac{n^{10}}{16 m^6}$

  Now let's examine Option B: $\frac{n^{10}}{16 m^6}$. Hmm, this looks promising! We have $n^{10}$ in the numerator, just like in our simplified expression. And we also have $16 m^6$ in the denominator, which perfectly matches our result. This is like finding that puzzle piece that slides right into place. Option B matches our simplified expression exactly. It looks like we might have found our winner!

• Option C: $\frac{n^{10}}{8 m^8}$

  Let's take a peek at Option C: $\frac{n^{10}}{8 m^8}$. At first glance, the $n^{10}$ in the numerator looks good. But when we look at the denominator, we see $8 m^8$. In our simplified expression, the denominator is $16 m^6$. The coefficient (the number in front of the 'm') is different, and the exponent on the 'm' is also different. So, Option C is not equivalent to our simplified expression. Nice try, Option C, but no cigar!

• Option D: $\frac{4 m^3}{n^8}$

  Finally, let's consider Option D: $\frac{4 m^3}{n^8}$. This one looks quite different from our simplified expression. We have 'm' in the numerator and 'n' in the denominator, which is the opposite of what we have in our expression. The exponents also don't match up at all. So, Option D is definitely not equivalent to our simplified expression. It’s important to check every part of the expression, guys.

The Winning Option

After carefully comparing each option to our simplified expression, it's clear that Option B, $\frac{n^{10}}{16 m^6}$, is the winner! It matches our simplified form perfectly, making it the equivalent expression. This is such a satisfying moment – like cracking a really tough code!

Conclusion: Option B is the Equivalent Expression

So, to wrap it all up, we started with the expression $(\frac{4 m n}{m^{-2} n^6})^{-2}$, and after some serious simplifying using the rules of exponents, we landed on $\frac{n^{10}}{16 m^6}$. We then compared this simplified expression with the options provided and found that Option B, $\frac{n^{10}}{16 m^6}$, is the equivalent expression. You nailed it! Keep practicing, and these types of problems will become second nature. You got this!

Remember, the key to these problems is to take it step by step, apply the rules of exponents carefully, and double-check your work along the way. Math is like a puzzle, and every step you take brings you closer to the solution. Keep up the awesome work, and you'll be a math whiz in no time!