Simplifying The Expression ((3xy^-5)^3 / (x^-2y^2)^-4)^-2 A Step-by-Step Solution

This article aims to provide a comprehensive, step-by-step solution to the mathematical expression: $\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$. We will meticulously break down each step, ensuring clarity and understanding for readers of all backgrounds. This exploration delves into the fundamental principles of exponent manipulation, offering valuable insights into simplifying complex algebraic expressions. Mastering these principles is crucial for success in various mathematical domains, from basic algebra to advanced calculus.

Understanding the Expression

The expression we aim to simplify is $\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$. It appears complex at first glance, but it's essentially a combination of terms involving variables $x$ and $y$ raised to various powers. Our goal is to apply the rules of exponents systematically to reduce this expression to its simplest equivalent form. This simplification involves dealing with negative exponents, powers of products, powers of quotients, and nested exponents. Each step is governed by established mathematical rules, which we will highlight as we proceed. By the end of this guide, you will not only understand the solution but also the underlying principles that make it possible. This knowledge will empower you to tackle similar problems with confidence and accuracy.

Step 1: Applying the Power of a Product Rule

In this initial step, we focus on simplifying the numerator of the fraction inside the parentheses. The numerator is $\left(3 x y^{-5}\right)^3$. To simplify this, we apply the power of a product rule, which states that $(ab)^n = a^n b^n$. This rule essentially distributes the exponent outside the parentheses to each factor inside. Applying this rule, we get $(3 x y^{-5})^3 = 3^3 x^3 (y^{-5})^3$. Next, we simplify each term individually. $3^3$ equals 27, and $(y^{-5})^3$ is simplified using the power of a power rule, which we'll discuss in the next step. So, after this step, our expression becomes $\left(\frac{27 x^3 y^{-15}}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$. This transformation brings us closer to simplifying the overall expression. Understanding the power of a product rule is fundamental for simplifying expressions involving exponents, and this step demonstrates its practical application.

Step 2: Applying the Power of a Power Rule

Now, we need to simplify $(y^{-5})^3$ in the numerator and $(x^{-2} y^2)^{-4}$ in the denominator. This requires the power of a power rule, which states that $(a^m)^n = a^{mn}$. This rule tells us that when we raise a power to another power, we multiply the exponents. Applying this to $(y^{-5})^3$, we get $y^{-5 \cdot 3} = y^{-15}$. For the denominator, we first apply the power of a product rule again: $(x^{-2} y^2)^{-4} = (x^{-2})^{-4} (y^2)^{-4}$. Then, applying the power of a power rule to each term, we get $(x^{-2})^{-4} = x^{-2 \cdot -4} = x^8$ and $(y^2)^{-4} = y^{2 \cdot -4} = y^{-8}$. So, the denominator simplifies to $x^8 y^{-8}$. Now, our expression looks like $\left(\frac{27 x^3 y^{-15}}{x^8 y^{-8}}\right)^{-2}$. This step highlights the importance of the power of a power rule in manipulating exponents. By consistently applying this rule, we can systematically simplify complex expressions.

Step 3: Simplifying the Fraction Inside the Parentheses

At this stage, we focus on simplifying the fraction $\frac{27 x^3 y^{-15}}{x^8 y^{-8}}$. To do this, we use the quotient of powers rule, which states that $\frac{a^m}{a^n} = a^{m-n}$. Applying this rule to the $x$ terms, we get $\frac{x^3}{x^8} = x^{3-8} = x^{-5}$. For the $y$ terms, we have $\frac{y^{-15}}{y^{-8}} = y^{-15 - (-8)} = y^{-15 + 8} = y^{-7}$. Therefore, the fraction simplifies to $27 x^{-5} y^{-7}$. Now, the entire expression becomes $\left(27 x^{-5} y^{-7}\right)^{-2}$. This simplification significantly reduces the complexity of the expression, making it easier to handle in the next steps. The quotient of powers rule is a key tool in simplifying expressions involving fractions with exponents.

Step 4: Applying the Power of a Product Rule Again

We now have the expression $\left(27 x^{-5} y^{-7}\right)^{-2}$. We apply the power of a product rule again, distributing the exponent -2 to each term inside the parentheses: $(27 x^{-5} y^{-7})^{-2} = 27^{-2} (x^{-5})^{-2} (y^{-7})^{-2}$. This step prepares us to simplify each term individually using the power of a power rule. Applying the power of a product rule strategically allows us to break down complex expressions into smaller, more manageable parts. This approach is particularly useful when dealing with multiple variables and exponents.

Step 5: Applying the Power of a Power Rule One Last Time

In this final simplification step, we apply the power of a power rule to each term. We have $27^{-2}$, $(x^{-5})^{-2}$, and $(y^{-7})^{-2}$. Let's simplify each one: $27^{-2} = \frac{1}{27^2} = \frac{1}{729}$. $(x^{-5})^{-2} = x^{-5 \cdot -2} = x^{10}$. $(y^{-7})^{-2} = y^{-7 \cdot -2} = y^{14}$. Combining these results, we get $\frac{1}{729} x^{10} y^{14}$, which can be written as $\frac{x^{10} y^{14}}{729}$. This completes the simplification process. By consistently applying the power of a power rule, we have successfully reduced the expression to its simplest form. This final step demonstrates the power and elegance of exponent rules in simplifying algebraic expressions.

Final Answer

Therefore, the expression $\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$ is equivalent to $\frac{x^{10} y^{14}}{729}$. So, the correct answer is A. $\frac{x^{10} y^{14}}{729}$. This result underscores the importance of mastering exponent rules for simplifying complex algebraic expressions. By following a systematic approach and applying the rules correctly, we can confidently tackle even the most challenging problems.

Summary of Key Concepts

Throughout this detailed solution, we've employed several key concepts related to exponents. These concepts are crucial for simplifying algebraic expressions and are worth summarizing for clarity:

1. Power of a Product Rule: $(ab)^n = a^n b^n$. This rule allows us to distribute an exponent over a product.
2. Power of a Power Rule: $(a^m)^n = a^{mn}$. This rule dictates how to handle exponents raised to other exponents.
3. Quotient of Powers Rule: $\frac{a^m}{a^n} = a^{m-n}$. This rule helps simplify fractions with exponents.
4. Negative Exponents: $a^{-n} = \frac{1}{a^n}$. Understanding negative exponents is essential for simplifying expressions.

By mastering these rules and practicing their application, you can confidently simplify a wide range of exponential expressions. The journey through this problem serves as a testament to the power and elegance of these fundamental mathematical principles. As you continue your mathematical exploration, remember that a solid foundation in these concepts will pave the way for success in more advanced topics.