Simplify Cube Root Expression Of 686x^4y^7
In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to present complex mathematical statements in a more concise and manageable form. This article delves into the process of simplifying the expression $\sqrt[3]{686 x^4 y^7}$, providing a step-by-step guide to arrive at the most simplified form. We will explore the underlying principles of radicals and exponents, and apply them to break down the given expression. Understanding these concepts is crucial for success in algebra and beyond. Let's embark on this journey of simplification together, unraveling the intricacies of radicals and exponents.
Understanding Radicals and Exponents
Before we dive into the simplification process, it's essential to grasp the fundamental concepts of radicals and exponents. Radicals, often represented by the radical symbol (), indicate the root of a number. For instance, represents the square root of 9, which is 3. Similarly, represents the cube root of 8, which is 2. The small number above the radical symbol, known as the index, indicates the type of root to be taken. When the index is not explicitly written, it is understood to be 2, representing the square root. Exponents, on the other hand, represent repeated multiplication of a base number. For example, means x multiplied by itself four times. Understanding the relationship between radicals and exponents is crucial for simplifying expressions involving them. Radicals can be expressed as fractional exponents, and vice versa, which allows us to manipulate expressions more easily. For instance, can be written as . This equivalence is a powerful tool in simplifying complex expressions. Mastering these concepts will pave the way for tackling more challenging mathematical problems with confidence and precision.
Step-by-Step Simplification of $\sqrt[3]{686 x^4 y^7}$
Now, let's embark on the journey of simplifying the expression $\sqrt[3]{686 x^4 y^7}$. This process involves breaking down the expression into its prime factors and then applying the properties of radicals and exponents. Our goal is to extract any perfect cube factors from under the radical, leaving the remaining factors within the radical. This systematic approach ensures that we arrive at the most simplified form of the expression. Each step in this process is crucial, building upon the previous one to gradually reduce the complexity of the expression. By following these steps carefully, we can transform the seemingly daunting expression into a more manageable and understandable form. Let's begin by examining the numerical coefficient, 686, and determining its prime factorization.
1. Prime Factorization of 686
To begin simplifying the expression, we first need to find the prime factorization of 686. This involves breaking down 686 into its prime factors, which are the prime numbers that multiply together to give 686. By systematically dividing 686 by prime numbers, we can identify these factors. Prime factorization is a fundamental tool in simplifying radicals, as it allows us to identify perfect cube factors that can be extracted from under the radical. The process of prime factorization not only helps in simplifying radicals but also provides a deeper understanding of the number's structure. This understanding is invaluable in various mathematical contexts. Let's proceed with the prime factorization of 686, unraveling its constituent prime numbers.
686 can be divided by 2, giving us 343. 343 can be divided by 7, giving us 49. 49 is 7 squared (7 * 7). Therefore, the prime factorization of 686 is 2 * 7 * 7 * 7, or 2 * 73.
2. Rewriting the Expression
Now that we have the prime factorization of 686, we can rewrite the original expression $\sqrt[3]{686 x^4 y^7}$ using this information. Replacing 686 with its prime factors, we get $\sqrt[3]{2 * 7^3 * x^4 * y^7}$. This step is crucial as it sets the stage for extracting perfect cube factors from under the radical. By expressing the numerical coefficient in terms of its prime factors, we can clearly identify which factors can be taken out of the cube root. Similarly, we will rewrite the variables with exponents as products of terms with exponents that are multiples of 3, along with any remaining factors. This rearrangement allows us to apply the properties of radicals and exponents effectively. Rewriting the expression in this way is a key step towards simplifying it, making the subsequent steps more straightforward.
3. Separating the Radicals
Next, we will separate the radical expression into a product of individual radicals, each containing a single factor. This is based on the property that $\sqrt[n]{a * b} = \sqrt[n]{a} * \sqrt[n]{b}$. Applying this property to our expression, we get $\sqrt[3]{2 * 7^3 * x^4 * y^7} = \sqrt[3]{2} * \sqrt[3]{7^3} * \sqrt[3]{x^4} * \sqrt[3]{y^7}$. Separating the radicals allows us to focus on simplifying each factor individually. This approach makes the simplification process more manageable and less prone to errors. By breaking down the complex radical into simpler components, we can apply the properties of radicals and exponents more effectively. This separation is a crucial step in isolating the perfect cube factors and extracting them from under the radical. Let's proceed with further simplification of each individual radical.
4. Simplifying Individual Radicals
Now, let's simplify each individual radical term. We'll start with the numerical terms and then move on to the variables. For $\sqrt[3]{7^3}$, the cube root of 73 is simply 7, as the cube root and the cube power cancel each other out. Next, we need to address the variable terms. For $\sqrt[3]{x^4}$, we can rewrite x4 as x3 * x. Similarly, for $\sqrt[3]{y^7}$, we can rewrite y7 as y6 * y, which is the same as (y2)3 * y. This rewriting allows us to identify perfect cube factors within the radicals. The ability to manipulate exponents and rewrite terms in this way is crucial for simplifying radical expressions. By extracting the perfect cube factors, we can reduce the complexity of the expression and arrive at its simplified form. Let's continue with the simplification process, applying these principles to each individual radical.
5. Extracting Perfect Cubes
Now, we extract the perfect cubes from under the radicals. For $\sqrt[3]{x^4}$, we have $\sqrt[3]{x^3 * x}$, which simplifies to x$\sqrt[3]{x}$. Similarly, for $\sqrt[3]{y^7}$, we have $\sqrt[3]{(y2)3 * y}$, which simplifies to y2$\sqrt[3]{y}$. This step is where the core simplification occurs, as we remove the factors that have a cube root, leaving the remaining factors under the radical. The ability to identify and extract perfect cubes is essential for simplifying radical expressions. By carefully examining the exponents and rewriting the terms, we can effectively remove the perfect cubes and reduce the complexity of the expression. This process brings us closer to the final simplified form. Let's proceed with combining the simplified terms and presenting the final answer.
6. Combining Simplified Terms
Finally, we combine the simplified terms. We have 7 from $\sqrt[3]{7^3}$, x from $\sqrt[3]{x^4}$, and y2 from $\sqrt[3]{y^7}$. Multiplying these terms together, we get 7xy2. The remaining terms under the radicals are $\sqrt[3]{2}$, $\sqrt[3]{x}$, and $\sqrt[3]{y}$. Combining these under a single radical, we get $\sqrt[3]{2xy}$. Therefore, the simplified expression is 7xy2$\sqrt[3]{2xy}$. This is the final step in our simplification journey, where we bring together all the individual simplified components to form the complete answer. The result is a more concise and manageable form of the original expression, making it easier to work with in further mathematical operations. The ability to simplify expressions like this is a crucial skill in algebra and beyond, enabling us to solve more complex problems with greater efficiency and accuracy. Let's reflect on the entire simplification process and appreciate the power of these fundamental mathematical principles.
Final Simplified Expression
After meticulously following the steps of prime factorization, rewriting the expression, separating the radicals, simplifying individual radicals, and extracting perfect cubes, we have successfully simplified the expression $\sqrt[3]{686 x^4 y^7}$. The final simplified form is 7xy2$\sqrt[3]{2xy}$. This result showcases the power of breaking down complex problems into smaller, manageable steps. By understanding the properties of radicals and exponents, we can effectively manipulate expressions and arrive at their simplest forms. This skill is not only essential for success in mathematics but also develops critical thinking and problem-solving abilities that are valuable in various aspects of life. The journey of simplifying this expression has provided a clear demonstration of how fundamental mathematical principles can be applied to unravel complexity and reveal underlying simplicity.
Conclusion
In conclusion, simplifying expressions like $\sqrt[3]{686 x^4 y^7}$ requires a solid understanding of radicals, exponents, and prime factorization. By systematically applying these concepts, we can transform complex mathematical statements into more manageable forms. The process involves breaking down the expression into its prime factors, identifying perfect cube factors, extracting them from under the radical, and combining the remaining terms. This step-by-step approach ensures that we arrive at the most simplified form of the expression. The ability to simplify expressions is a fundamental skill in mathematics, enabling us to solve equations, analyze functions, and tackle more advanced mathematical concepts. Furthermore, the problem-solving skills developed through this process are transferable to other areas of life, fostering critical thinking and analytical abilities. Mastering these techniques empowers us to approach mathematical challenges with confidence and precision. Remember, practice is key to mastering these skills, so continue to explore and simplify various expressions to solidify your understanding and enhance your mathematical prowess. The journey of mathematical exploration is a continuous one, and each simplified expression brings us one step closer to a deeper understanding of the mathematical world.
