Simplifying Cube Root Expressions: A Step-by-Step Guide

Simplifying radical expressions, especially cube roots, can seem daunting at first. But with a clear understanding of the underlying principles and a systematic approach, you can master the simplification of any cube root expression. This guide will walk you through the process of simplifying the expression $\sqrt[3]{686 x^4 y^7}$, breaking it down into manageable steps and providing explanations along the way. By the end of this guide, you'll not only be able to simplify this specific expression but also have the tools and knowledge to tackle similar problems.

Understanding Cube Roots

Before diving into the simplification process, it's crucial to understand what a cube root represents. The cube root of a number a, denoted as $\sqrt[3]{a}$, is a value that, when multiplied by itself three times, equals a. In other words, if $\sqrt[3]{a} = b$, then b * b* * b* = a. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Similarly, the cube root of 27 is 3 because 3 * 3 * 3 = 27. This understanding forms the foundation for simplifying expressions involving cube roots.

When dealing with variables inside a cube root, we look for powers that are multiples of 3. This is because $\sqrt[3]{x^3} = x$, $\sqrt[3]{x^6} = x^2$, and so on. If the power of a variable is not a multiple of 3, we can rewrite it as a product of a power that is a multiple of 3 and a power that is less than 3. For instance, $x^4$ can be written as $x^3 * x$, and $y^7$ can be written as $y^6 * y$. This technique is essential for extracting variables from the cube root.

Prime Factorization

Prime factorization is a fundamental tool in simplifying radical expressions, including cube roots. It involves breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. For example, the prime factorization of 12 is 2 * 2 * 3, often written as $2^2 * 3$. Similarly, the prime factorization of 36 is 2 * 2 * 3 * 3, or $2^2 * 3^2$. Prime factorization helps us identify perfect cubes within the radicand (the number inside the cube root symbol), making it easier to simplify the expression. For example, the prime factorization of 8 is 2 * 2 * 2, or $2^3$, which directly shows that the cube root of 8 is 2.

To find the prime factorization of a number, you can use a factor tree or successive division by prime numbers. Start by dividing the number by the smallest prime number that divides it evenly (usually 2, 3, 5, 7, etc.). Then, continue factoring the resulting quotients until you are left with only prime numbers. The prime factorization of 686, for instance, will reveal a perfect cube factor, which is crucial for simplifying the given expression $\sqrt[3]{686 x^4 y^7}$. Understanding prime factorization is key to unlocking the simplification of numerical parts of cube root expressions.

Step-by-Step Simplification of $\sqrt[3]{686 x^4 y^7}$

Let's break down the simplification of the cube root expression $\sqrt[3]{686 x^4 y^7}$ into a series of clear, manageable steps:

Step 1: Prime Factorization of the Coefficient

Begin by finding the prime factorization of the numerical coefficient, 686. This will help identify any perfect cube factors. 686 can be factored as 2 * 343. Further factoring 343, we find that it is 7 * 49, and 49 is 7 * 7. Therefore, the prime factorization of 686 is 2 * 7 * 7 * 7, which can be written as 2 * $7^3$. This reveals that $7^3$ is a perfect cube factor within 686.

Step 2: Rewrite the Expression Using Prime Factors

Now, rewrite the original expression using the prime factorization of 686: $\sqrt[3]{686 x^4 y^7} = \sqrt[3]{2 * 7^3 * x^4 * y^7}$. This step makes it easier to see the perfect cubes and other factors that can be extracted from the cube root.

Step 3: Separate Variables with Powers as Multiples of 3

Next, focus on the variable terms, $x^4$ and $y^7$. Rewrite these terms as products of powers that are multiples of 3 and the remaining factors. $x^4$ can be written as $x^3 * x$, and $y^7$ can be written as $y^6 * y$. So, the expression becomes $\sqrt[3]{2 * 7^3 * x^3 * x * y^6 * y}$.

Step 4: Apply the Cube Root to Perfect Cubes

Apply the cube root to the perfect cube factors. Remember that $\sqrt[3]{a * b} = \sqrt[3]{a} * \sqrt[3]{b}$. In this case, we have $\sqrt[3]{7^3} = 7$, $\sqrt[3]{x^3} = x$, and $\sqrt[3]{y^6} = y^2$ (since $y^6 = (y^2)^3$). Extract these perfect cubes from the cube root: $7 * x * y^2 * \sqrt[3]{2 * x * y}$.

Step 5: Simplify the Expression

Finally, simplify the expression by combining the terms outside the cube root: $7 x y^2 \sqrt[3]{2 x y}$. This is the simplified form of the original expression. By following these steps systematically, you can confidently simplify cube root expressions.

Detailed Breakdown of Variable Simplification

When simplifying cube root expressions, dealing with variables requires a specific approach. The key is to rewrite the variables' exponents in terms of multiples of 3, because the cube root of a variable raised to a power of 3 is the variable itself. Let's delve deeper into how we handle variables like $x^4$ and $y^7$ in the expression $\sqrt[3]{686 x^4 y^7}$.

For $x^4$, we aim to rewrite it as a product of a power that is a multiple of 3 and a remaining factor. Since 4 = 3 + 1, we can write $x^4$ as $x^3 * x^1$, or simply $x^3 * x$. This separation allows us to take the cube root of $x^3$, which is x, and leave the remaining x inside the cube root. This technique is crucial for simplifying expressions where the exponent is not a multiple of 3.

Similarly, for $y^7$, we need to find the largest multiple of 3 that is less than or equal to 7. In this case, it is 6. So, we rewrite $y^7$ as $y^6 * y^1$, or $y^6 * y$. Since $y^6$ can be written as $(y^2)^3$, its cube root is $y^2$. The remaining y stays inside the cube root. This method ensures that we extract the largest possible perfect cube factors from the variables, simplifying the overall expression.

By breaking down the variables in this manner, we can effectively simplify any variable term within a cube root expression. This step-by-step approach ensures that we accurately extract the cube roots of the variable components, leading to the fully simplified form of the expression.

Final Answer: $7 x y^2 \sqrt[3]{2 x y}$

After meticulously following the steps of prime factorization, variable separation, and cube root extraction, we arrive at the simplified form of the expression $\sqrt[3]{686 x^4 y^7}$. The process involved breaking down the coefficient 686 into its prime factors, which revealed the perfect cube $7^3$. We then rewrote the variable terms $x^4$ and $y^7$ as products of powers that are multiples of 3 and the remaining factors. This allowed us to extract the cube roots of $7^3$, $x^3$, and $y^6$, leaving the remaining factors inside the cube root.

By extracting these perfect cube factors, we obtained the terms 7, x, and $y^2$ outside the cube root. The remaining factors inside the cube root were 2, x, and y. Combining these terms, we arrive at the final simplified expression: $7 x y^2 \sqrt[3]{2 x y}$. This result matches option B from the multiple-choice options, confirming that this is the correct simplified form of the given expression.

This step-by-step simplification not only provides the answer but also offers a clear understanding of the process involved. By mastering these techniques, you can confidently tackle similar problems involving cube roots and radical expressions.

Conclusion

Simplifying cube root expressions, such as $\sqrt[3]{686 x^4 y^7}$, requires a systematic approach that involves prime factorization, variable manipulation, and careful extraction of perfect cube factors. By breaking down the expression into manageable steps, we can identify and extract the cube roots of numerical coefficients and variable terms, leading to the simplified form. This process not only provides the solution but also enhances our understanding of radical expressions and their properties.

Mastering these simplification techniques is crucial for success in algebra and higher-level mathematics. The ability to confidently simplify expressions allows for easier manipulation and problem-solving in various mathematical contexts. Remember to practice these steps with different examples to solidify your understanding and improve your skills. With consistent practice, simplifying cube root expressions will become second nature.

For further exploration and practice on simplifying radical expressions, consider visiting Khan Academy's Algebra I section, which offers a wealth of resources and exercises to enhance your understanding.