Simplify Cube Root Expressions A Step-by-Step Guide

In the fascinating realm of mathematics, simplifying radical expressions often presents a captivating challenge. This article delves into the intricacies of summing cube roots, specifically addressing the expression $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$. We will embark on a step-by-step journey, unraveling the layers of this expression and ultimately arriving at its simplified form. This exploration will not only enhance your understanding of radical operations but also equip you with the tools to tackle similar mathematical puzzles. Our journey begins with a careful examination of the properties of cube roots and exponents, paving the way for a systematic simplification process. Let's dive into the world of cube roots and discover the elegance hidden within this expression. By the end of this comprehensive guide, you will have a clear understanding of how to approach and solve such problems, adding a valuable skill to your mathematical arsenal. Remember, the beauty of mathematics lies not just in the answers, but also in the process of discovery and understanding. So, let's embark on this mathematical adventure together, unlocking the secrets of cube root simplification.

Understanding Cube Roots and Exponents

Before we embark on the simplification journey, it's crucial to solidify our understanding of cube roots and exponents. A cube root of a number is a value that, when multiplied by itself three times, yields the original number. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8. In mathematical notation, the cube root is represented by the symbol $\sqrt[3]{}$. Exponents, on the other hand, indicate the number of times a base is multiplied by itself. For example, $x^{3}$ means x multiplied by itself three times (x * x * x). When dealing with exponents within radicals, we can leverage the property that $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. This property is fundamental to simplifying radical expressions. In the context of cube roots, this translates to $\sqrt[3]{a^m} = a^{\frac{m}{3}}$. Grasping this concept is paramount as it allows us to convert radical expressions into exponential forms, making them easier to manipulate and simplify. Furthermore, understanding how exponents interact with multiplication is essential. The rule $(ab)^n = a^n b^n$ allows us to distribute an exponent over a product, which is a powerful tool when simplifying expressions under a radical. With these fundamental concepts in hand, we are well-prepared to tackle the expression at hand. The journey of simplification often involves breaking down complex expressions into smaller, more manageable parts. By understanding the interplay between cube roots and exponents, we can confidently navigate the simplification process and arrive at the solution.

Breaking Down the Expression

Now, let's dissect the given expression: $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$. The first step in simplifying this expression is to break down each term under the cube root into its prime factors and variable components. This approach allows us to identify perfect cubes within the radicals, which can then be extracted. Starting with the first term, $\sqrt[3]{125 x^{10} y^{13}}$, we recognize that 125 is a perfect cube (5 * 5 * 5 = 125). For the variable components, we have $x^{10}$ and $y^{13}$. To extract terms from the cube root, we need to find the largest multiples of 3 that are less than or equal to the exponents. For $x^{10}$, the largest multiple of 3 is 9, so we can rewrite it as $x^9 * x$. Similarly, for $y^{13}$, the largest multiple of 3 is 12, allowing us to rewrite it as $y^{12} * y$. Applying the same logic to the second term, $\sqrt[3]{27 x^{10} y^{13}}$, we identify 27 as a perfect cube (3 * 3 * 3 = 27). The variable components remain the same, $x^{10}$ and $y^{13}$, which we can decompose as before. By breaking down the expression in this manner, we are essentially preparing the ground for extracting perfect cube factors from the radicals. This process is akin to peeling away the layers of an onion, revealing the simpler components within. With this decomposition complete, we are one step closer to simplifying the expression and revealing its true form. The next stage involves extracting the perfect cube factors and simplifying the remaining terms.

Extracting Perfect Cubes

Having broken down the expression, our next endeavor is to extract the perfect cubes from each term. Recall that $\sqrt[3]{125 x^{10} y^{13}}$ can be rewritten as $\sqrt[3]{5^3 * x^9 * x * y^{12} * y}$. Similarly, $\sqrt[3]{27 x^{10} y^{13}}$ can be expressed as $\sqrt[3]{3^3 * x^9 * x * y^{12} * y}$. Now, we can apply the property $\sqrt[3]{a * b} = \sqrt[3]{a} * \sqrt[3]{b}$ to separate the perfect cubes from the remaining factors. This yields: $\sqrt[3]{5^3} * \sqrt[3]{x^9} * \sqrt[3]{x} * \sqrt[3]{y^{12}} * \sqrt[3]{y}$ for the first term and $\sqrt[3]{3^3} * \sqrt[3]{x^9} * \sqrt[3]{x} * \sqrt[3]{y^{12}} * \sqrt[3]{y}$ for the second term. Now, we simplify the cube roots of the perfect cubes. $\sqrt[3]{5^3} = 5$, $\sqrt[3]{x^9} = x^3$ (since 9 / 3 = 3), and $\sqrt[3]{y^{12}} = y^4$ (since 12 / 3 = 4). Similarly, $\sqrt[3]{3^3} = 3$. Substituting these values back into the expression, we get: $5x^3y^4\sqrt[3]{xy} + 3x^3y^4\sqrt[3]{xy}$. This step is crucial as it transforms the expression into a form where we can clearly see the common factors. By extracting the perfect cubes, we have effectively simplified the radicals, making the subsequent steps much more straightforward. The process of extraction is akin to separating the wheat from the chaff, isolating the components that can be readily simplified. With the perfect cubes extracted, we are now poised to combine like terms and arrive at the final simplified expression.

Combining Like Terms

With the perfect cubes extracted, we've arrived at a crucial juncture: combining like terms. Our expression now stands as $5x^3y^4\sqrt[3]{xy} + 3x^3y^4\sqrt[3]{xy}$. Notice that both terms share a common factor: $x^3y^4\sqrt[3]{xy}$. This common factor allows us to treat the terms as similar entities, making them eligible for addition. Just as we can combine 5 apples and 3 apples to get 8 apples, we can combine these terms by adding their coefficients. The coefficients in this case are 5 and 3. Adding these coefficients, we get 5 + 3 = 8. Therefore, we can rewrite the expression as $(5 + 3)x^3y^4\sqrt[3]{xy}$. Performing the addition, we obtain $8x^3y^4\sqrt[3]{xy}$. This is the simplified form of the original expression. By combining like terms, we have effectively reduced the expression to its most concise and elegant form. This step highlights the importance of recognizing common factors in mathematical expressions. It's akin to finding the common thread in a tapestry, allowing us to weave the individual strands into a unified whole. The ability to combine like terms is a fundamental skill in algebra, and it is essential for simplifying expressions and solving equations. With this final simplification, we have successfully navigated the complexities of the original expression and arrived at a clear and concise solution. The journey from the initial cube root expression to its simplified form demonstrates the power of mathematical manipulation and the beauty of algebraic simplification.

Final Simplified Expression

After meticulously breaking down, extracting, and combining, we arrive at the final simplified expression: $8x^3y^4\sqrt[3]{xy}$. This expression represents the culmination of our step-by-step journey through the intricacies of cube root simplification. It encapsulates the essence of the original expression, $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$, in a more concise and manageable form. The simplified expression not only provides a clear answer but also showcases the power of algebraic manipulation and the elegance of mathematical simplification. It allows for easier interpretation and further manipulation if needed. This final result underscores the importance of understanding the underlying principles of cube roots, exponents, and factoring. It demonstrates how these concepts can be applied to transform complex expressions into simpler, more understandable forms. The journey to this simplified expression has been a testament to the systematic approach to problem-solving in mathematics. By breaking down the problem into smaller, manageable steps, we have successfully navigated the complexities of the original expression. The final simplified expression serves as a testament to the power of mathematical techniques and the beauty of algebraic simplification. It is a concise and elegant representation of the original expression, ready for further use or interpretation.

Simplify the sum of the cube roots: $\sqrt[3]{125 x^{10} y^{13}}+\sqrt[3]{27 x^{10} y^{13}}$

Simplify Cube Root Expressions A Step-by-Step Guide