Simplifying Radical Expressions Finding The Sum Of 2(³√16x³y) + 4(³√54x⁶y⁵)

In this article, we embark on a mathematical journey to unravel the intricacies of the following sum: $2\left(\sqrt[3]{16 x^3 y}\right)+4\left(\sqrt[3]{54 x^6 y^5}\right)$. This expression, at first glance, might appear daunting, but with a systematic approach and a solid understanding of radical simplification, we can break it down into manageable components and arrive at a concise solution. Our exploration will involve leveraging the properties of radicals, factoring out perfect cubes, and combining like terms. By the end of this discussion, you will not only understand the solution but also gain valuable insights into the techniques for handling similar mathematical expressions. This problem serves as an excellent exercise in algebraic manipulation and reinforces the fundamental principles of working with radicals. So, let's dive in and dissect this fascinating sum step by step.

Understanding the Fundamentals of Radicals

Before we tackle the main problem, it's crucial to establish a firm grasp on the fundamental properties of radicals, particularly cube roots. A radical is a mathematical expression that involves a root, such as a square root, cube root, or nth root. In our case, we are dealing with cube roots, denoted by the symbol $\sqrt[3]{}$. The cube root of a number a is a value that, when multiplied by itself three times, equals a. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Understanding this basic concept is pivotal for simplifying expressions involving radicals.

When dealing with expressions under a radical, we can utilize the property that the cube root of a product is equal to the product of the cube roots. Mathematically, this is expressed as $\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$. This property allows us to break down complex radicals into simpler terms by factoring out perfect cubes. A perfect cube is a number that can be obtained by cubing an integer. Examples of perfect cubes include 1 (1^3), 8 (2^3), 27 (3^3), 64 (4^3), and so on. Identifying and extracting perfect cubes from under the radical is a key step in simplifying radical expressions.

Furthermore, when variables are involved under the radical, we can apply similar principles. For example, $\sqrt[3]{x^3} = x$ because x multiplied by itself three times is x³. In general, $\sqrt[3]{x^{3n}} = x^n$, where n is an integer. This understanding will be crucial when we simplify the given expression, which involves variables raised to different powers under the cube root. By mastering these fundamental concepts, we lay a solid foundation for effectively tackling the problem at hand.

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

Let's begin by dissecting the first term of our expression: $2\left(\sqrt[3]{16 x^3 y}\right)$. Our primary goal here is to simplify the cube root portion, $\sqrt[3]{16 x^3 y}$, by identifying and extracting any perfect cubes. We start by examining the numerical coefficient, 16. We need to determine if 16 has any factors that are perfect cubes. The prime factorization of 16 is 2 * 2 * 2 * 2, which can be written as 2⁴. We can rewrite this as 2³ * 2, where 2³ is a perfect cube (2³ = 8). Therefore, we can express 16 as 8 * 2.

Now, let's consider the variable part of the expression, $x^3 y$. The term $x^3$ is already a perfect cube since it's x raised to the power of 3. The term y, however, is not a perfect cube as it is raised to the power of 1. With this analysis, we can rewrite the cube root portion as follows:

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

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

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

Now we can simplify the cube roots of the perfect cubes. We know that $\sqrt[3]{8} = 2$ and $\sqrt[3]{x^3} = x$. Substituting these values, we have:

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

Combining these terms, we get:

$2x\sqrt[3]{2y}$

Finally, we multiply this simplified cube root by the coefficient 2 from the original term: $2\left(\sqrt[3]{16 x^3 y}\right) = 2 \cdot 2x\sqrt[3]{2y} = 4x\sqrt[3]{2y}$. This completes the simplification of the first term, providing us with a more manageable expression to work with.

Simplifying 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, our objective is to simplify the cube root portion, $\sqrt[3]{54 x^6 y^5}$, by identifying and extracting perfect cubes. We begin by examining the numerical coefficient, 54. The prime factorization of 54 is 2 * 3 * 3 * 3, which can be written as 2 * 3³. Here, we observe that 3³ is a perfect cube (3³ = 27). Therefore, we can express 54 as 27 * 2.

Next, we turn our attention to the variable part of the expression, $x^6 y^5$. The term $x^6$ can be seen as a perfect cube since the exponent 6 is divisible by 3. Specifically, $x^6 = (x^2)^3$. The term $y^5$ is not a perfect cube in its current form, but we can rewrite it as $y^3 \cdot y^2$, where $y^3$ is a perfect cube. With this analysis, we can rewrite the cube root portion as follows:

$\sqrt[3]{54 x^6 y^5} = \sqrt[3]{27 \cdot 2 \cdot x^6 \cdot y^3 \cdot y^2}$

Now, we apply the property of radicals that allows us to separate the cube root of a product into the product of cube roots:

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

We can now simplify the cube roots of the perfect cubes. We know that $\sqrt[3]{27} = 3$, $\sqrt[3]{x^6} = x^2$, and $\sqrt[3]{y^3} = y$. Substituting these values, we have:

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

Combining these terms, we get:

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

Finally, we multiply this simplified cube root by the coefficient 4 from the original term: $4\left(\sqrt[3]{54 x^6 y^5}\right) = 4 \cdot 3x^2y\sqrt[3]{2y^2} = 12x^2y\sqrt[3]{2y^2}$. This completes the simplification of the second term, providing us with another manageable expression.

Combining the Simplified Terms and Final Solution

Now that we have successfully simplified both terms of the original expression, we can combine them to find the final solution. Recall that the simplified first term is $4x\sqrt[3]{2y}$ and the simplified second term is $12x^2y\sqrt[3]{2y^2}$. Our original expression was:

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

Substituting the simplified terms, we get:

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

At this point, we need to examine the terms to see if there are any further simplifications we can make. Specifically, we look for like terms, which are terms that have the same radical part and the same variables raised to the same powers. In this case, we have two terms:

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

Upon closer inspection, we observe that the radical parts, $\sqrt[3]{2y}$ and $\sqrt[3]{2y^2}$, are different, and the variables have different powers as well. The first term has x to the power of 1 and y inside the cube root, while the second term has x² and y outside the cube root, and y² inside the cube root. Because the terms are not alike, we cannot combine them further.

Therefore, the final simplified expression is:

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

This is the sum of the simplified radicals, and it represents the most concise form of the original expression. Through this step-by-step simplification process, we have successfully tackled a complex radical expression and arrived at a clear and understandable solution. This exercise highlights the importance of understanding the properties of radicals and applying them systematically to solve mathematical problems.

In conclusion, we have successfully navigated the intricacies of the sum $2\left(\sqrt[3]{16 x^3 y}\right)+4\left(\sqrt[3]{54 x^6 y^5}\right)$, arriving at the simplified expression $4x\sqrt[3]{2y} + 12x^2y\sqrt[3]{2y^2}$. This journey involved a thorough understanding of radical properties, strategic factoring to identify perfect cubes, and careful simplification of each term. The process underscored the importance of breaking down complex problems into smaller, more manageable steps. By applying the principles of radical simplification, we were able to transform an initially daunting expression into a clear and concise form. This exercise not only provides a solution to the specific problem but also reinforces fundamental algebraic techniques applicable to a wide range of mathematical challenges. The ability to manipulate radical expressions is a valuable skill in mathematics, and the step-by-step approach demonstrated here can serve as a template for tackling similar problems in the future. This exploration highlights the beauty and power of mathematical simplification, showcasing how complex expressions can be tamed through methodical application of core principles.