Simplifying Expressions With Positive Exponents

Hey everyone! Today, we're diving into a fun math problem that's all about simplifying expressions using only positive exponents. We'll break down a complex expression step by step, making sure you understand every bit of it. Plus, we'll fill in those blanks like pros. Let's get started, shall we?

The Problem: Unpacking the Expression

Our expression is: $\left[\frac{\left(x^2 y^3\right)^{-1}}{\left(x^{-2} y^2 z\right)^2}\right]^2$, with the conditions $x \neq 0$, $y \neq 0$, and $z \neq 0$. This might look a bit intimidating at first glance, but trust me, we'll make it super manageable. The key here is to carefully apply the rules of exponents. We are going to simplify this expression by applying the power of a product rule, the power of a quotient rule, and the negative exponent rule. It's like a puzzle, and we're the solvers! The main goal is to get everything into a form where all the exponents are positive, which is super important in algebra and other areas of math. So, let's get those exponent rules ready and start simplifying the equation.

First, let's clarify the rules we'll be using. The power of a product rule says that if you have something like $(ab)^n$, it simplifies to $a^n b^n$. Next up is the power of a quotient rule, which means that $(\frac{a}{b})^n$ equals $\frac{a^n}{b^n}$. Last but not least is the negative exponent rule. This means $a^{-n}$ is the same as $\frac{1}{a^n}$. Keeping these rules in mind is crucial as we simplify the given expression, and it will give us an easy time solving the equations. Now, we're going to use these rules to simplify this complicated-looking expression into something simpler. Remember, the goal is to get rid of negative exponents and make everything positive and easier to read. Always pay close attention to the order of operations, and then it is a walk in the park! We'll start with the inner parts of the fraction and work our way outwards. This approach helps us to stay organized and not miss any important steps along the way. So, buckle up! We are going to make it simple.

Step 1: Handling the Inner Powers

Alright, let's start with the numerator and the denominator separately. This is where we use the power of a product rule and the negative exponent rule. First, we tackle the numerator, which is $(x^2 y^3)^{-1}$. Applying the power rule to each term, we get $x^{2*(-1)} y^{3*(-1)}$. This simplifies to $x^{-2} y^{-3}$. Great job, guys!

Now, let's move to the denominator. We have $(x^{-2} y^2 z)^2$. Using the power rule again, we get $x^{-2*2} y^{2*2} z^{1*2}$, which simplifies to $x^{-4} y^4 z^2$. Keep up the good work! We're doing great.

So far, we've taken the first steps to making our expression simpler. We've applied the power rule to simplify the exponents within the parentheses. This step gets rid of the outer exponent on the terms, making the overall equation much simpler to work with. It's like peeling back the layers of an onion – each step reveals a simpler form. Remember to focus on each part of the expression individually to avoid making mistakes. Consistency is key here. Now that we've broken down the numerator and the denominator, we're ready to combine them and move on to the next phase of simplification. You're doing great, and we're halfway there.

Step 2: Combining and Simplifying the Fraction

Now we're going to put the numerator and the denominator back together, so our expression looks like this: $\frac{x^{-2} y^{-3}}{x^{-4} y^4 z^2}$. Next, we're going to use the quotient rule of exponents. This involves subtracting the exponents of the same base. Let's take the x's first. We have $x^{-2 - (-4)}$, which becomes $x^{-2 + 4}$, simplifying to $x^2$. Then, for the y's, we have $y^{-3 - 4}$, which becomes $y^{-7}$. And finally, the $z^2$ stays as is since there is no other z in the numerator. Now we have $x^2 y^{-7} z^{-2}$ in the denominator. You are all doing amazing, and keep up the great work!

This step brings us closer to having an expression with positive exponents. We've seen how to combine the terms in the numerator and denominator using the quotient rule, resulting in a single term for each variable. Now, our expression looks simpler and more organized, and it’s easier to see the next steps. Using the quotient rule is a critical skill for simplifying expressions. Remember, the rule involves subtracting the exponents, which simplifies the terms. We are getting very close to our goal. Let's see how we can make the exponents positive and complete the process. This phase is important to avoid mistakes when simplifying the exponents, and it requires our full attention to correctly manage and combine terms. Keep your eyes on the prize and focus on each variable individually. Once you get the hang of it, simplifying these expressions becomes much easier.

Step 3: Dealing with Negative Exponents

We're almost there, folks! Now, we have $x^2 y^{-7} z^{-2}$. To get rid of those pesky negative exponents, we use the negative exponent rule, which means we move the terms with negative exponents to the denominator. This turns our expression into $\frac{x^2}{y^7 z^2}$. We did it! All the exponents are positive, and the expression is simplified.

Here, we use the negative exponent rule to transform the expression so that it has only positive exponents. This rule is simple but essential. By moving the terms, we flipped the sign of the exponents, which simplified the expression. It is like a final touch to bring everything to its simplest form. Now the expression is ready for us to use in any mathematical scenario. Having positive exponents makes the expression easier to interpret. It's easier to use in calculations and in understanding what's going on. This final step is all about making the answer clear and easily understandable. Congratulations on reaching this point! Now, we're ready to complete the statements.

Step 4: Applying the Outer Power

Now, let's go back to the original expression: $\left[\frac{x^{-2} y^{-3}}{x^{-4} y^4 z^2}\right]^2$. Before we simplified it to $\frac{x^2}{y^7 z^2}$, we can also apply the outer power of 2 from the start. This means we square each term inside the brackets. So, we'll apply the exponent 2 to everything, $x^2 y^{-7} z^{-2}$.

Applying the power to each term inside the fraction, we get $\frac{(x^2)^2}{(y^7)^2 (z^2)^2}$, which simplifies to $\frac{x^4}{y^{14} z^4}$. And there you have it, folks! The final, simplified form, ready to solve the last part of our problem. We are pros now!

Completing the Statements

Now, let's address the specific question. We need to find the exponent on $x$ in the simplified expression. Looking at our final simplified expression, $\frac{x^4}{y^{14} z^4}$, we see that the exponent on $x$ is 4. So, the answer is:

The exponent on $x$ is $\boxed{4}$.

Conclusion: You Did It!

And that's a wrap, everyone! We've successfully simplified the expression and found the exponent on $x$. Remember, the key is to break down the problem step by step, apply the rules of exponents correctly, and be patient. Keep practicing, and you'll become a pro at simplifying expressions in no time. If you got stuck at any point, go back and review the rules. Math can be fun when you have the right approach. Awesome job, and keep up the great work! If you have any questions or want to try another problem, feel free to ask. See ya!