Simplifying Algebraic Expressions A Step-by-Step Solution

Hey guys! Today, we're going to dive deep into the world of algebraic expressions and tackle a particularly interesting problem. We'll be working to find an equivalent expression for the somewhat intimidating expression: $\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}$. Sounds complex, right? But don't worry, we'll break it down step by step, making sure everyone understands the logic and the math behind it. Algebraic expressions might seem daunting at first, especially when they involve negative exponents and multiple variables. But trust me, with a systematic approach and a clear understanding of the rules of exponents, you can conquer any such problem. This kind of question is crucial not only for acing your math exams but also for building a solid foundation for more advanced mathematical concepts. So, let's roll up our sleeves and get started on this exciting journey of simplifying this algebraic puzzle!

Deciphering the Expression The Key to Simplifying Complex Algebraic Equations

Okay, let's start by dissecting the expression and understanding its components. Our main expression is $\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}$. The first thing we notice is the presence of nested exponents and negative exponents. Remember, guys, negative exponents indicate reciprocals. For instance, $x^{-n}$ is the same as $\frac{1}{x^n}$. This is a fundamental rule that we'll use extensively. We also have the power of a power rule, which states that $(x^m)^n = x^{mn}$. This means when we raise a power to another power, we multiply the exponents. Next, we'll deal with the exponents outside the parentheses. Squaring the numerator means we apply the exponent 2 to each factor inside the parentheses. Similarly, the exponent -2 in the denominator applies to both $3$, $a^5$, and $b$. The final exponent of -1 outside the entire fraction means we'll eventually need to take the reciprocal of the entire simplified expression. Understanding these exponent rules is absolutely crucial for simplifying this expression. Without a solid grasp of these rules, we'd be lost in a sea of variables and exponents. So, before we proceed, make sure you're comfortable with these basic principles. We're building a strong foundation here, and each step depends on the one before it. Now, let's move on to the next stage where we'll apply these rules to simplify the expression.

Applying Exponent Rules Step-by-Step Simplification for Clarity

Now, let's put those exponent rules into action! We'll start by simplifying the numerator and the denominator separately. For the numerator, we have $\left(2 a^{-3} b^4\right)^2$. Applying the power of a power rule, we get $2^2 \cdot a^{-3 \cdot 2} \cdot b^{4 \cdot 2}$, which simplifies to $4a^{-6}b^8$. Remember, we're multiplying the exponents inside the parentheses by the exponent outside. This is a critical step, so make sure you're following along. For the denominator, we have $\left(3 a^5 b\right)^{-2}$. Applying the same rule, we get $3^{-2} \cdot a^{5 \cdot (-2)} \cdot b^{-2}$, which simplifies to $3^{-2}a^{-10}b^{-2}$. Notice how the negative exponent outside the parentheses affects the exponents of each term inside. This is a common area for mistakes, so pay close attention. Now, let's rewrite the original expression with these simplified terms: $\left(\frac{4a^{-6}b^8}{3^{-2}a^{-10}b^{-2}}\right)^{-1}$. We've made significant progress, but we're not done yet! The next step is to deal with the fraction inside the parentheses. When dividing terms with the same base, we subtract the exponents. So, we'll subtract the exponents of $a$ and $b$ in the denominator from those in the numerator. This will further simplify the expression and bring us closer to our final answer. Remember, the key here is to take it one step at a time, applying the exponent rules systematically. We're not rushing, we're making sure each step is clear and correct. Let's move on to the next step and see how this subtraction of exponents plays out.

Simplifying the Fraction Combining Terms with Precision

Alright, guys, let's tackle that fraction! We've got $\frac{4a^{-6}b^8}{3^{-2}a^{-10}b^{-2}}$. Remember, when dividing terms with the same base, we subtract the exponents. So, for the $a$ terms, we have $a^{-6} / a^{-10}$, which becomes $a^{-6 - (-10)} = a^{-6 + 10} = a^4$. Notice how subtracting a negative exponent becomes addition. This is a crucial detail! For the $b$ terms, we have $b^8 / b^{-2}$, which becomes $b^{8 - (-2)} = b^{8 + 2} = b^{10}$. Again, watch out for those negative signs! Now, let's deal with the constants. We have $4$ in the numerator and $3^{-2}$ in the denominator. Remember that $3^{-2}$ is the same as $\frac{1}{3^2} = \frac{1}{9}$. So, we have $\frac{4}{\frac{1}{9}}$. To divide by a fraction, we multiply by its reciprocal, so this becomes $4 \cdot 9 = 36$. Putting it all together, the fraction simplifies to $36a^4b^{10}$. We've made a huge leap in simplifying our expression! We've combined the $a$ terms, the $b$ terms, and the constants, all while carefully applying the rules of exponents. But remember, we still have that exponent of -1 outside the parentheses. This is the final piece of the puzzle, and it's going to flip our expression one last time. So, let's move on to the final step and see how this negative exponent transforms our result.

The Final Flip Applying the Last Exponent for the Solution

Okay, guys, we're in the home stretch! We've simplified the expression inside the parentheses to $36a^4b^{10}$. Now we need to apply the exponent of -1, which means we need to take the reciprocal of the entire expression. So, $\left(36a^4b^{10}\right)^{-1}$ becomes $\frac{1}{36a^4b^{10}}$. And there you have it! We've successfully simplified the original complex expression to a much cleaner and more manageable form. We started with a tangle of exponents and fractions, and through careful application of exponent rules and step-by-step simplification, we've arrived at our final answer. This process highlights the power of breaking down complex problems into smaller, more manageable steps. Each step, from dealing with negative exponents to combining like terms, built upon the previous one, leading us to the solution. Remember, guys, math isn't about memorizing formulas, it's about understanding the underlying principles and applying them logically. We've done exactly that in this problem. We've used the rules of exponents as our tools, and we've navigated through the complexities of the expression with clarity and precision. So, let's take a moment to appreciate the journey we've taken, from the initial intimidating expression to the final elegant solution. Now, let's look at the answer choices and see which one matches our result.

Matching the Solution Identifying the Correct Equivalent Expression

Now that we've simplified the expression to $\frac{1}{36 a^4 b^{10}}$, let's compare it to the given answer choices. Our options are:

A. $\frac{2}{3 a^4 b^{10}}$ B. $\frac{4}{9 a^4 b^{10}}$ C. $\frac{1}{36 a^4 b^{10}}$ D. \frac{36 a^4 b^{10}}

By simply comparing our simplified expression with the options, we can clearly see that option C, $\frac{1}{36 a^4 b^{10}}$, matches our result perfectly. So, C is the correct answer! Guys, this is a great feeling, isn't it? We started with a challenging problem, and through our hard work and understanding of the concepts, we've arrived at the correct solution. This process of matching our result with the answer choices is a crucial step in problem-solving. It's a way to double-check our work and ensure that we haven't made any mistakes along the way. It also reinforces our understanding of the problem and the solution. So, always take the time to compare your final answer with the given options. It's a simple step that can make a big difference in your accuracy. Now, let's take a moment to recap the entire process and reinforce the key concepts we've learned.

Recap and Key Takeaways Mastering Exponent Rules for Algebraic Success

Alright, guys, let's recap what we've learned in this awesome journey of simplifying algebraic expressions! We started with the expression $\left(\frac{\left(2 a^{-3} b^4\right)^2}{\left(3 a^5 b\right)^{-2}}\right)^{-1}$ and our mission was to find an equivalent expression. We tackled this problem by systematically applying the rules of exponents. First, we identified the key rules we needed: the power of a power rule, the rule for negative exponents, and the rule for dividing terms with the same base. Then, we broke down the expression step by step. We simplified the numerator and denominator separately, carefully applying the power of a power rule. We dealt with the negative exponents, remembering that they indicate reciprocals. We combined like terms by subtracting exponents when dividing. Finally, we applied the outermost exponent of -1, which meant taking the reciprocal of the entire expression. Through this process, we arrived at the simplified expression $\frac{1}{36 a^4 b^{10}}$, which matched option C in the answer choices. The key takeaways from this exercise are:

• Mastering exponent rules: A solid understanding of exponent rules is crucial for simplifying algebraic expressions.
• Breaking down complex problems: Complex problems can be solved by breaking them down into smaller, more manageable steps.
• Systematic approach: A systematic approach, applying rules one at a time, leads to accurate solutions.
• Double-checking your work: Always compare your final answer with the given options to ensure accuracy.

Guys, I hope this detailed explanation has helped you understand how to simplify complex algebraic expressions. Remember, practice is key! The more you work with these types of problems, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep unlocking the mysteries of mathematics! We've conquered this challenge together, and I'm super proud of your effort! Now, go out there and tackle some more algebraic puzzles!