Simplify The Expression: A Math Problem Solved!

Are you ready to dive into a fascinating math problem? We're going to break down a complex expression step by step, making it super easy to understand. This isn't just about getting the right answer; it's about learning the fundamental principles of algebra and how to apply them. So, grab your thinking cap, and let's get started!

Understanding the Problem

Our mission, should we choose to accept it, is to simplify the following expression:

$\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}$

Where we assume that $a \neq 0$ and $b \neq 0$. This might look intimidating at first glance, but don't worry! We'll tackle it piece by piece. The key here is to remember the rules of exponents. Exponents are your friends in this mathematical journey. We'll use these rules to make this expression much simpler. First, let’s identify the core concepts we'll be using:

• Power of a Power: $(x^m)^n = x^{m*n}$
• Product to a Power: $(xy)^n = x^n y^n$
• Negative Exponents: $x^{-n} = \frac{1}{x^n}$
• Quotient to a Power: $(\frac{x}{y})^n = \frac{x^n}{y^n}$

With these rules in our arsenal, we're ready to conquer this problem! We will apply these concepts diligently, breaking down each part of the expression until we arrive at a simplified form. Remember, mathematics is like building a structure; each step is a brick, and with solid foundations, we can construct something beautiful and logical. It's not about memorizing; it's about understanding the 'why' behind each step. Let's begin this exciting journey!

Breaking Down the Expression: Step-by-Step

Let's start by simplifying the numerator and the denominator separately. This will make the whole process much more manageable. Remember, we're like detectives solving a mystery – each clue (or step) leads us closer to the solution. The first part we'll focus on is the numerator:

Simplifying the Numerator: $\left(2 a^{-3} b^4\right)^2$

Using the "Product to a Power" rule, we distribute the exponent 2 to each term inside the parentheses:

$\left(2 a^{-3} b^4\right)^2 = 2^2 * (a^{-3})^2 * (b^4)^2$

Now, applying the "Power of a Power" rule:

$2^2 * (a^{-3})^2 * (b^4)^2 = 4 * a^{-6} * b^8$

So, the simplified numerator is $4 a^{-6} b^8$. See? Not so scary when we break it down. Each part of the problem is like a puzzle piece, and now we've neatly fitted one piece into place. We've successfully navigated the exponents and coefficients in the numerator, paving the way for our next step. By taking it slowly and methodically, we avoid errors and gain a clearer understanding of the process.

Simplifying the Denominator: $\left(3 a^5 b\right)^{-2}$

Similarly, we apply the "Product to a Power" rule and distribute the exponent -2:

$\left(3 a^5 b\right)^{-2} = 3^{-2} * (a^5)^{-2} * b^{-2}$

Using the "Power of a Power" rule again:

$3^{-2} * (a^5)^{-2} * b^{-2} = 3^{-2} * a^{-10} * b^{-2}$

And using the "Negative Exponents" rule, we rewrite $3^{-2}$ as $\frac{1}{3^2} = \frac{1}{9}$:

$\frac{1}{9} * a^{-10} * b^{-2}$

So, the simplified denominator is $\frac{1}{9} a^{-10} b^{-2}$. We're making great progress! The expression is slowly transforming from something complex into a more manageable form. Just like an artist chiseling away at a sculpture, we are refining our expression, revealing its underlying simplicity. Now that we've tackled both the numerator and the denominator, we're ready to combine them and see what happens next.

Combining the Numerator and Denominator

Now we have:

$\frac{4 a^{-6} b^8}{\frac{1}{9} a^{-10} b^{-2}}$

Dividing by a fraction is the same as multiplying by its reciprocal, so:

$\frac{4 a^{-6} b^8}{\frac{1}{9} a^{-10} b^{-2}} = 4 a^{-6} b^8 * \frac{9}{a^{-10} b^{-2}}$

Which gives us:

$36 a^{-6} b^8 * a^{10} b^{2}$

When multiplying terms with the same base, we add the exponents:

$36 * a^{-6+10} * b^{8+2} = 36 a^4 b^{10}$

We're almost there! The expression is becoming increasingly streamlined. Think of it as decluttering a room – we're getting rid of the unnecessary complexity and revealing the clean, simple core. By combining the numerator and denominator, we've consolidated our terms and are now just a step away from the final simplification.

The Final Step: Applying the Outer Exponent

Remember, the entire expression was raised to the power of -1:

$\left(36 a^4 b^{10}\right)^{-1}$

Applying the "Product to a Power" and "Negative Exponents" rules:

$\left(36 a^4 b^{10}\right)^{-1} = 36^{-1} * (a^4)^{-1} * (b^{10})^{-1} = \frac{1}{36} * a^{-4} * b^{-10}$

Rewriting with positive exponents:

$\frac{1}{36 a^4 b^{10}}$

The Answer

Therefore, the equivalent expression is:

$\frac{1}{36 a^4 b^{10}}$

So, the correct answer is C. $\frac{1}{36 a^4 b^{10}}$. Congratulations, mathletes! We've successfully navigated a complex algebraic expression and emerged victorious. This journey wasn't just about finding the right answer; it was about understanding the underlying principles, mastering the rules of exponents, and learning to approach problems with a step-by-step strategy. By breaking down the problem into manageable parts, we turned a daunting task into a series of smaller, achievable goals.

Conclusion

We've successfully simplified the given expression by applying the rules of exponents methodically. Remember, the key to solving complex problems is to break them down into smaller, more manageable steps. This approach not only makes the problem less intimidating but also helps in understanding the underlying concepts better. Keep practicing, and you'll become a math whiz in no time! You've now equipped yourself with the tools and knowledge to tackle similar challenges with confidence. Remember, mathematics is a journey of discovery, and each problem solved is a step forward. Keep exploring, keep questioning, and keep learning!

For further exploration of exponent rules and practice problems, visit Khan Academy's Exponents and Radicals Section. This resource offers comprehensive lessons and exercises to reinforce your understanding of these concepts.