Simplifying Cube Root Of (x^3 * Y^12) / 64 Completely Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of simplifying the cube root of the expression ${\sqrt[3]{\frac{x^3 y^{12}}{64}}}$ completely. This problem falls under the domain of algebra and involves the simplification of radicals, specifically cube roots, and the application of exponent rules. Mastering such simplifications is crucial for success in various mathematical disciplines, including calculus, differential equations, and advanced algebra. This article aims to provide a step-by-step solution, ensuring clarity and understanding for students and enthusiasts alike. Whether you are a high school student preparing for an exam or someone looking to refresh your algebra skills, this guide will offer valuable insights and techniques to tackle similar problems effectively.

Understanding Cube Roots and Radicals

Before we dive into the simplification, let's establish a solid foundation by understanding cube roots and radicals. A cube root of a number is a value that, when multiplied by itself three times, yields the original number. For example, the cube root of 8 is 2 because ${2 \times 2 \times 2 = 8}$. Mathematically, the cube root of a number ${a}$ is denoted as ${\sqrt[3]{a}}$. Radicals, in general, are expressions that involve roots, such as square roots, cube roots, and higher-order roots. Simplifying radicals involves expressing them in their simplest form, where the radicand (the number inside the root) has no perfect cube factors (in the case of cube roots). The ability to manipulate and simplify radicals is a fundamental skill in algebra, playing a pivotal role in solving equations, simplifying expressions, and tackling complex mathematical problems. A strong grasp of radical simplification not only enhances problem-solving speed but also fosters a deeper understanding of mathematical structures and relationships. Therefore, mastering the concepts of cube roots and radicals is essential for any student aiming for proficiency in mathematics.

Step-by-Step Simplification Process

To simplify the expression ${\sqrt[3]{\frac{x^3 y^{12}}{64}}}$ completely, we will follow a step-by-step approach. This methodical process will not only lead us to the correct answer but also illustrate the underlying principles of simplifying radicals. Each step is designed to break down the problem into manageable parts, ensuring clarity and reducing the chance of errors. Understanding each step thoroughly will equip you with the skills to tackle similar problems with confidence.

Step 1: Apply the Quotient Rule for Radicals

The first key step in simplifying our expression is to apply the quotient rule for radicals. This rule states that the cube root of a fraction is equal to the fraction of the cube roots of the numerator and the denominator. In mathematical terms:

${ \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} }$

Applying this rule to our expression, we get:

${ \sqrt[3]{\frac{x^3 y^{12}}{64}} = \frac{\sqrt[3]{x^3 y^{12}}}{\sqrt[3]{64}} }$

This separation allows us to deal with the numerator and the denominator independently, simplifying the overall process. The quotient rule is a fundamental tool in radical simplification, making complex expressions more manageable and easier to understand. By applying this rule, we lay the groundwork for the subsequent steps in our simplification journey. This initial separation is crucial as it allows us to focus on individual components, ensuring a more organized and efficient simplification process.

Step 2: Simplify the Cube Root of the Numerator

Now, let's focus on simplifying the cube root of the numerator, which is (\sqrt[3]{x^3 y^{12}}. To simplify this, we can use the property that the cube root of a product is the product of the cube roots. In other words:

${ \sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b} }$

Applying this property, we can rewrite the cube root of the numerator as:

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

Next, we simplify each cube root individually. The cube root of ${x^3}$ is simply ${x}$, because ${x \cdot x \cdot x = x^3}$. For {\sqrt[3]{y^{12}}\, we need to find a power of \(y} that, when cubed, equals ${y^{12}}$. We can find this by dividing the exponent 12 by 3:

${ \frac{12}{3} = 4 }$

So, ${\sqrt[3]{y^{12}} = y^4}$, because ${(y^4)^3 = y^{12}}$. Therefore, the simplified form of the cube root of the numerator is:

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

This step demonstrates how understanding the properties of radicals and exponents can lead to significant simplifications. By breaking down the numerator into its constituent parts and applying the appropriate rules, we were able to express it in a much simpler form. This simplification is a critical component of solving the overall problem and showcases the power of algebraic manipulation.

Step 3: Simplify the Cube Root of the Denominator

In this step, we will simplify the cube root of the denominator, which is ${\sqrt[3]{64}}$. To find the cube root of 64, we need to identify a number that, when multiplied by itself three times, equals 64. We can express 64 as a product of its prime factors to make this easier:

${ 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 }$

Now, we want to find a number ${n}$ such that ${n^3 = 64}$. In terms of the prime factorization, we are looking for a number whose cube is ${2^6}$. We can rewrite ${2^6}$ as ${(2^2)^3}$, which simplifies to ${4^3}$. Therefore:

${ \sqrt[3]{64} = \sqrt[3]{4^3} = 4 }$

Alternatively, you might recognize that ${4 \times 4 \times 4 = 64}$ directly. The ability to quickly identify perfect cubes is a valuable skill in simplifying radicals. This simplification of the denominator is crucial for the final answer, as it provides a concrete numerical value that can be combined with the simplified numerator. Understanding how to find cube roots of numbers, whether through prime factorization or direct recognition, is an essential skill in algebra.

Step 4: Combine the Simplified Numerator and Denominator

Now that we have simplified both the numerator and the denominator, we can combine them to obtain the final simplified expression. From Step 2, we found that the cube root of the numerator is ${xy^4}$, and from Step 3, we found that the cube root of the denominator is 4. Therefore, we can write the simplified expression as:

${ \frac{\sqrt[3]{x^3 y^{12}}}{\sqrt[3]{64}} = \frac{xy^4}{4} }$

This fraction represents the completely simplified form of the original expression. There are no further simplifications possible, as the numerator and denominator have no common factors other than 1. The process of combining the simplified parts is a crucial step in any simplification problem, as it brings together all the individual components into a cohesive final answer. This step highlights the importance of accuracy in the previous steps, as any errors would propagate to the final result. By correctly combining the simplified numerator and denominator, we arrive at the complete solution to our problem.

Final Simplified Expression

After following the step-by-step process, we have successfully simplified the original expression. The final simplified expression is:

${ \frac{xy^4}{4} }$

This is the most reduced form of the given radical expression. It is essential to present the final answer clearly, as it represents the culmination of all the simplification efforts. This final result demonstrates the power of algebraic manipulation and the importance of understanding the properties of radicals and exponents. By breaking down the problem into smaller, manageable steps, we were able to simplify a complex expression into a concise and elegant form. This process not only provides the answer but also reinforces the principles of mathematical simplification, equipping students with the tools to tackle similar problems effectively.

Conclusion

In conclusion, simplifying radical expressions such as ${\sqrt[3]{\frac{x^3 y^{12}}{64}}}$ requires a strong understanding of radical properties and exponent rules. By applying the quotient rule for radicals, simplifying cube roots of individual terms, and combining the results, we arrived at the completely simplified expression ${\frac{xy^4}{4}}$. This step-by-step guide not only provides the solution but also illustrates the methodology behind simplifying complex algebraic expressions. Mastering these techniques is crucial for success in higher-level mathematics. The ability to simplify radicals efficiently and accurately is a valuable skill that will benefit students in various mathematical contexts. By consistently practicing these techniques, students can develop a deeper understanding of algebraic principles and enhance their problem-solving capabilities. This comprehensive guide serves as a valuable resource for students and enthusiasts alike, empowering them to tackle radical simplification problems with confidence and precision.