Simplifying Expressions: Eliminating Negative Exponents

Hey math enthusiasts! Today, we're diving into the world of simplifying expressions and tackling those pesky negative exponents. Let's break down the expression $rac{x y^{-6}}{x{-4} y^2}$ and figure out how to rewrite it without any negative exponents. It's like a little algebra adventure, and I'm here to guide you through it. This topic is super important in mathematics because it lays the groundwork for more complex equations and problem-solving. Understanding how exponents work, especially when they're negative, gives you a solid base for calculus, physics, and even computer science. So, buckle up; it's going to be a fun ride! We'll go through each step, ensuring you understand the 'why' behind every move. By the end of this, you'll be a pro at eliminating negative exponents and simplifying expressions like a boss. Let's get started!

So, the original problem is $rac{x y^{-6}}{x{-4} y^2}$, and we are asked to find the correct answer from the following options:

A. $rac{x^4}{y2 x^6 y^6}$ B. $rac{x x^4}{y2 y^6}$ C. $rac{x^4}{y2 x}$

Let's get into it. First, remember that a negative exponent means we need to move the term to the opposite side of the fraction bar. For instance, $y^{-6}$ is the same as rac{1}{y^6}, and $x^{-4}$ is the same as rac{1}{x^4}. When you see a negative exponent, think of it as a signal to flip its position in the fraction! Got it? Now, let's make some moves. Here's how we can simplify it step-by-step.

Step-by-Step Simplification

Okay, guys, the first thing we need to do is address those negative exponents. We have $y^{-6}$ and $x^{-4}$. Let's tackle them one by one. Remember the rule: a^{-n} = rac{1}{a^n}.

Addressing the Negative Exponents

1. Dealing with $y^{-6}$: Since $y^{-6}$ is in the numerator, we move it down to the denominator and change the exponent's sign. So, $y^{-6}$ becomes rac{1}{y^6}.
2. Dealing with $x^{-4}$: Since $x^{-4}$ is in the denominator, we move it up to the numerator, and the exponent's sign changes. So, $x^{-4}$ becomes $x^4$.

Now, the expression looks like this: $rac{x imes x^4}{y2 imes y^6}$. See how we've gotten rid of those pesky negative exponents? Great job!

Simplifying the Expression

Now that we've dealt with the negative exponents, it's time to simplify the expression further. Let's combine like terms. When multiplying terms with the same base, we add their exponents. Let's apply this rule to the numerator and denominator.

1. Combine the x terms: In the numerator, we have $x$ (which is $x^1$) and $x^4$. When we multiply these, we get $x^{1+4} = x^5$.
2. Combine the y terms: In the denominator, we have $y^2$ and $y^6$. When we multiply these, we get $y^{2+6} = y^8$.

So, our simplified expression is: $rac{x^5}{y8}$. This is the final, simplified form of the expression without negative exponents. We started with $rac{x y^{-6}}{x{-4} y^2}$ and transformed it step by step. Now, we have a much cleaner and easier-to-understand expression. Remember the key takeaways: move terms with negative exponents to the opposite side of the fraction, and combine like terms by adding their exponents. You are doing awesome!

Evaluating the Options

Alright, now that we have successfully simplified the original expression $rac{x y^{-6}}{x{-4} y^2}$ to $rac{x^5}{y8}$, let's check the options provided. Our job now is to find the option that matches our simplified answer. This is a crucial step in any math problem because it helps us make sure our calculations are correct. By comparing our final answer with the given options, we can confirm that we have followed all the rules of exponents correctly. Let's analyze each option one by one. Remember, we are looking for an expression that is equivalent to $rac{x^5}{y8}$. Pay close attention, as this is where we apply our understanding. So, let's see if any of the options are correct.

Option A: $rac{x^4}{y2 x^6 y^6}$

This option looks a bit messy, doesn't it? Let's see if we can simplify it. First, combine the x terms in the denominator: $x^6$. Then, combine the y terms in the denominator: $y^2 imes y^6 = y^8$. So, this simplifies to $rac{x^4}{x6 y^8}$. Further simplification will lead to $rac{1}{x^2 y^8}$. This does not match $rac{x^5}{y8}$, so it's not the correct answer.

Option B: $rac{x x^4}{y2 y^6}$

This looks promising! Let's simplify it. In the numerator, we have $x imes x^4 = x^5$ (remember, we add the exponents when multiplying). In the denominator, we have $y^2 imes y^6 = y^8$. So, this simplifies to $rac{x^5}{y8}$. This exactly matches our simplified answer! Bingo!

Option C: $rac{x^4}{y2 x}$

Let's check this one. We can rewrite this as $racx^4}{x y^2}$. Now, we can simplify the x terms $rac{x^4{x} = x^3$. So, this simplifies to $rac{x^3}{y2}$. This does not match $rac{x^5}{y8}$, so it's not the correct answer.

Conclusion

Therefore, the correct answer is Option B: $rac{x x^4}{y2 y^6}$, which simplifies to $rac{x^5}{y8}$. We have successfully simplified the expression and found the correct match. Keep practicing these problems, and you'll become a pro in no time! Remember, the key is to be patient, follow the rules step-by-step, and always double-check your work. You got this! I hope this was helpful, guys. Keep practicing, and you will master negative exponents. Feel free to ask me more questions. Until next time, happy simplifying!