Simplifying Exponential Expressions: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of exponential expressions, specifically tackling a problem that seems a bit intimidating at first glance: $\left(\frac{5 x^5 y^3 z^{-2}}{10 x y^5 z^4}\right)^{-2}=\frac{A y^a z^b}{B x^c}$. Don't worry, it's not as scary as it looks. We'll break it down step by step, making sure you understand every move we make. The goal is to simplify this expression into a form where we can easily identify the values of A, a, B, b, and c. Are you ready? Let's get started!

Step 1: Simplify Inside the Parentheses

Our first step is to focus on the expression inside the parentheses: $\frac{5 x^5 y^3 z^{-2}}{10 x y^5 z^4}$. The key here is to simplify each part of the fraction separately.

Let's start with the coefficients (the numbers). We have $\frac{5}{10}$, which simplifies to $\frac{1}{2}$. Easy peasy! Now, let's move on to the variables. For the x terms, we have $\frac{x^5}{x}$. Remember that when you divide exponents with the same base, you subtract the powers. So, $x^5 / x^1$ becomes $x^{5-1} = x^4$. Next, the y terms: $\frac{y^3}{y^5}$. This becomes $y^{3-5} = y^{-2}$. Finally, the z terms: $\frac{z^{-2}}{z^4}$. This simplifies to $z^{-2-4} = z^{-6}$.

Putting it all together, the expression inside the parentheses simplifies to $\frac{1 x^4 y^{-2} z^{-6}}{2}$. We're making progress, guys! It's starting to look much cleaner. Remember, these little steps are what lead to the final answer. Keep your eyes on the prize, and let's keep simplifying this equation. Always focus on each part of the equation, as this is the key to solving the more complex problems. Make sure to understand what each exponent is doing to each term. The more you practice, the easier it will be to solve these types of equations. You will see that everything will become second nature! So, let's keep moving!

Step 2: Apply the Outer Exponent

Now that we've simplified the inside of the parentheses, we can address that pesky exponent of -2 on the outside: $\left(\frac{1 x^4 y^{-2} z^{-6}}{2}\right)^{-2}$. When you have an exponent raised to another exponent, you multiply the powers. And when you have a fraction raised to an exponent, you apply the exponent to both the numerator and the denominator.

So, let's apply the -2 exponent to each part of our simplified fraction. First, the numerator: $1^{-2} * (x^4)^{-2} * (y^{-2})^{-2} * (z^{-6})^{-2}$. Remember that anything raised to the power of -2 is the same as the reciprocal squared. Hence, $1^{-2} = 1$. Then, $(x^4)^{-2} = x^{-8}$, $(y^{-2})^{-2} = y^4$, and $(z^{-6})^{-2} = z^{12}$. Second, we apply the exponent to the denominator: $2^{-2} = \frac{1}{4}$.

So now we have $\frac{1 x^{-8} y^4 z^{12}}{\frac{1}{4}}$. To simplify this, we can rewrite the whole expression by dividing by $\frac{1}{4}$, which is the same as multiplying by 4. So now our equation is $\frac{4 y^4 z^{12}}{x^8}$.

Step 3: Identify A, a, B, b, and c

Almost there! Now, we compare our simplified expression, $\frac{4 y^4 z^{12}}{x^8}$, to the target form, $\frac{A y^a z^b}{B x^c}$. By matching the terms, we can easily see the values. We can find that A = 4, a = 4, B = 1, b = 12, and c = 8. Boom! We've successfully simplified the expression and identified all the missing values.

Final Answer

Let's recap what we've found:

• $A = 4$
• $a = 4$
• $B = 1$
• $b = 12$
• $c = 8$

Therefore, $\left(\frac{5 x^5 y^3 z^{-2}}{10 x y^5 z^4}\right)^{-2} = \frac{4 y^4 z^{12}}{x^8}$.

That was a fun one, right? We hope you learned something new today. Keep practicing, and you'll become a master of exponential expressions in no time! Keep in mind that understanding the properties of exponents is crucial for solving these types of problems. Remember the rules for dividing exponents, raising exponents to a power, and dealing with negative exponents. With these concepts in your toolkit, you'll be well-equipped to tackle any exponential expression that comes your way. And if you ever feel stuck, don't be afraid to break down the problem into smaller, more manageable steps, just like we did. You've got this!

Tips for Success

• Master the Basics: Make sure you're comfortable with the fundamental rules of exponents. Review the rules for multiplying, dividing, and raising exponents to powers. Understand what happens with negative exponents and how to simplify them.
• Break it Down: When faced with a complex expression, break it down into smaller, simpler steps. Simplify inside the parentheses first, then apply the outer exponent.
• Practice Regularly: The more you practice, the better you'll become. Work through different examples to build your confidence and understanding.
• Check Your Work: Always double-check your work to avoid silly mistakes. Go back and review each step to ensure accuracy.
• Ask for Help: Don't be afraid to ask for help if you get stuck. Your teacher, classmates, or online resources can provide valuable assistance.

Keep in mind that these are guidelines. Depending on your personal learning style, you may find that other methods or resources are more helpful. The most important thing is to find a strategy that works for you and to stick with it. Remember, practice makes perfect, so keep working at it, and you'll see your skills improve over time. Good luck, and happy simplifying!