Simplifying Radical Expressions A Step By Step Guide

$2\left(\sqrt[3]{16 x^3 y}\right)+4\left(\sqrt[3]{54 x^6 y^5}\right)$

Introduction: Embracing the Beauty of Algebraic Simplification

In the realm of mathematics, algebraic expressions often present themselves in intricate forms, challenging us to unveil their underlying simplicity. This exploration delves into the process of simplifying a radical expression, focusing on the sum of two terms involving cube roots. Our journey will involve extracting perfect cube factors, applying the properties of radicals, and combining like terms to arrive at the most concise representation of the given expression. Let's embark on this mathematical adventure, where we'll transform a seemingly complex expression into an elegant and understandable form. In this comprehensive guide, we'll break down each step, ensuring a clear understanding of the techniques involved in simplifying radical expressions. Get ready to delve into the world of algebraic manipulation, where we'll uncover the secrets behind simplifying radicals and algebraic expressions. We'll not only simplify the given expression but also equip you with the skills to tackle similar problems with confidence. So, let's begin this exciting journey of mathematical simplification, where we'll transform complexity into clarity and master the art of radical expression manipulation.

Deconstructing the First Term: $2\left(\sqrt[3]{16 x^3 y}\right)$

To begin our simplification journey, let's focus on the first term of the expression: $2\left(\sqrt[3]{16 x^3 y}\right)$. Our primary goal here is to identify and extract any perfect cube factors that reside within the cube root. By doing so, we can effectively reduce the complexity of the expression and pave the way for further simplification. The key to this process lies in recognizing that $16$ can be factored into $8 \cdot 2$, where $8$ is a perfect cube ($2^3$). Similarly, $x^3$ is already a perfect cube. With these observations, we can rewrite the term as follows:

$2\left(\sqrt[3]{16 x^3 y}\right) = 2\left(\sqrt[3]{8 \cdot 2 \cdot x^3 \cdot y}\right)$

Now, we can leverage the property of radicals that allows us to separate the cube root of a product into the product of cube roots: $\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$. Applying this property, we get:

$2\left(\sqrt[3]{8 \cdot 2 \cdot x^3 \cdot y}\right) = 2 \cdot \sqrt[3]{8} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{2y}$

We know that $\sqrt[3]{8} = 2$ and $\sqrt[3]{x^3} = x$, so we can substitute these values into the expression:

$2 \cdot \sqrt[3]{8} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{2y} = 2 \cdot 2 \cdot x \cdot \sqrt[3]{2y}$

Finally, multiplying the constants together, we arrive at the simplified form of the first term:

$2 \cdot 2 \cdot x \cdot \sqrt[3]{2y} = 4x\sqrt[3]{2y}$

This meticulous breakdown of the first term exemplifies the core principle of simplifying radicals: identifying and extracting perfect cube factors to reduce the expression to its most fundamental form. This process not only simplifies the expression but also lays the groundwork for combining like terms later on.

Deconstructing the Second Term: $4\left(\sqrt[3]{54 x^6 y^5}\right)$

Now, let's shift our focus to the second term of the expression: $4\left(\sqrt[3]{54 x^6 y^5}\right)$. Similar to our approach with the first term, we aim to identify and extract perfect cube factors from within the cube root. This process will help us simplify the term and potentially reveal common factors that can be combined with the simplified first term.

The first step is to factorize the coefficient, $54$. We can express $54$ as $27 \cdot 2$, where $27$ is a perfect cube ($3^3$). Next, we examine the variable terms. $x^6$ can be written as $(x^2)^3$, which is also a perfect cube. For $y^5$, we can express it as $y^3 \cdot y^2$, where $y^3$ is a perfect cube. With these factorizations in mind, we can rewrite the second term as:

$4\left(\sqrt[3]{54 x^6 y^5}\right) = 4\left(\sqrt[3]{27 \cdot 2 \cdot x^6 \cdot y^3 \cdot y^2}\right)$

Now, we apply the property of radicals that allows us to separate the cube root of a product into the product of cube roots: $\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$. This gives us:

$4\left(\sqrt[3]{27 \cdot 2 \cdot x^6 \cdot y^3 \cdot y^2}\right) = 4 \cdot \sqrt[3]{27} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^3} \cdot \sqrt[3]{2y^2}$

We know that $\sqrt[3]{27} = 3$, $\sqrt[3]{x^6} = x^2$, and $\sqrt[3]{y^3} = y$. Substituting these values into the expression, we get:

$4 \cdot \sqrt[3]{27} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^3} \cdot \sqrt[3]{2y^2} = 4 \cdot 3 \cdot x^2 \cdot y \cdot \sqrt[3]{2y^2}$

Finally, multiplying the constants together, we arrive at the simplified form of the second term:

$4 \cdot 3 \cdot x^2 \cdot y \cdot \sqrt[3]{2y^2} = 12x^2y\sqrt[3]{2y^2}$

By systematically identifying and extracting perfect cube factors, we've successfully simplified the second term. This process not only reduces the complexity of the term but also prepares us for the final step of combining the simplified terms.

Combining Like Terms: The Grand Finale

With both terms now simplified, we stand at the final stage of our journey: combining like terms. This step will reveal the ultimate simplified form of the original expression. Recall that we simplified the first term to $4x\sqrt[3]{2y}$ and the second term to $12x^2y\sqrt[3]{2y^2}$.

To combine terms, they must have the same radical part. In this case, we have $\sqrt[3]{2y}$ in the first term and $\sqrt[3]{2y^2}$ in the second term. Since the radicands (the expressions inside the cube roots) are different, these terms cannot be combined directly. This means that the simplified forms of the two terms, $4x\sqrt[3]{2y}$ and $12x^2y\sqrt[3]{2y^2}$, represent the most simplified expression we can achieve.

Therefore, the final simplified sum is:

$4x\sqrt[3]{2y} + 12x^2y\sqrt[3]{2y^2}$

This result underscores the importance of meticulous simplification and the recognition of like terms. While we couldn't combine the terms into a single term, we successfully reduced the original expression to its simplest form, revealing the inherent structure and relationships within the expression.

Conclusion: A Triumph of Simplification

In this mathematical endeavor, we've successfully navigated the complexities of simplifying a radical expression involving cube roots. We began with the sum $2\left(\sqrt[3]{16 x^3 y}\right)+4\left(\sqrt[3]{54 x^6 y^5}\right)$ and, through a systematic process of identifying and extracting perfect cube factors, arrived at the simplified form $4x\sqrt[3]{2y} + 12x^2y\sqrt[3]{2y^2}$.

Our journey involved several key steps:

1. Deconstructing the First Term: We factored $16$ into $8 \cdot 2$, extracted the perfect cube factor of $8$, and simplified the term to $4x\sqrt[3]{2y}$.
2. Deconstructing the Second Term: We factored $54$ into $27 \cdot 2$, $x^6$ into $(x^2)^3$, and $y^5$ into $y^3 \cdot y^2$. We then extracted the perfect cube factors and simplified the term to $12x^2y\sqrt[3]{2y^2}$.
3. Combining Like Terms: We recognized that the simplified terms did not have the same radical part and therefore could not be combined further.

This exploration highlights the power of algebraic manipulation and the importance of understanding the properties of radicals. By mastering these techniques, we can transform seemingly complex expressions into elegant and understandable forms. The final simplified expression, $4x\sqrt[3]{2y} + 12x^2y\sqrt[3]{2y^2}$, represents the culmination of our efforts, showcasing the beauty and precision of mathematical simplification. This journey not only provided a solution to the given problem but also reinforced the fundamental principles of algebraic manipulation and radical simplification, equipping us with the skills to tackle similar challenges in the future.

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