Simplifying Cube Roots Equivalent Expression For The Cube Root Of 256x^10y^7

In the realm of mathematics, simplifying expressions involving radicals and exponents is a fundamental skill. This article delves into the process of finding the equivalent expression for $\sqrt[3]{256 x^{10} y^7}$, providing a step-by-step solution and explaining the underlying concepts. Understanding how to manipulate radicals and exponents is crucial for success in algebra and beyond. We'll break down the expression, identify perfect cube factors, and extract them from the radical to arrive at the simplified form. This exploration will not only help you solve this specific problem but also equip you with the tools to tackle similar challenges in the future.

Deciphering the Expression: $\sqrt[3]{256 x^{10} y^7}$

The expression $\sqrt[3]{256 x^{10} y^7}$ represents the cube root of the product of 256, $x^{10}$, and $y^7$. To find an equivalent expression, we need to simplify the radical by identifying and extracting any perfect cube factors. A perfect cube is a number or expression that can be obtained by cubing an integer or algebraic term. For example, 8 is a perfect cube because $2^3 = 8$, and $x^3$ is a perfect cube because $(x)^3 = x^3$. Our goal is to rewrite the expression under the cube root as a product of perfect cubes and remaining factors, allowing us to simplify the radical.

The first step in simplifying $\sqrt[3]{256 x^{10} y^7}$ is to break down the expression into its individual components and identify perfect cube factors within each. We'll start by factoring the coefficient 256, then address the variable terms $x^{10}$ and $y^7$. Understanding the properties of exponents is essential for this process. Recall that $(a^m)^n = a^{m*n}$ and $\sqrt[n]{a^m} = a^{m/n}$. These rules will guide us in extracting perfect cubes from the radical.

Breaking Down 256

The number 256 can be factored into its prime factors. This process involves repeatedly dividing the number by its smallest prime factor until we reach 1. Factoring 256, we get: $256 = 2 \times 128 = 2 \times 2 \times 64 = 2 \times 2 \times 2 \times 32 = 2 \times 2 \times 2 \times 2 \times 16 = 2 \times 2 \times 2 \times 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$. Therefore, $256 = 2^8$. Now, to find the largest perfect cube factor of 256, we need to identify the highest multiple of 3 that is less than or equal to 8 (the exponent of 2). That multiple is 6, so we can rewrite $2^8$ as $2^6 \times 2^2$. Since $2^6 = (2^2)^3 = 4^3$, we have a perfect cube factor of $4^3 = 64$. The remaining factor is $2^2 = 4$. Thus, we can rewrite 256 as $64 \times 4$.

Simplifying $x^{10}$

Next, let's consider the variable term $x^{10}$. To extract a perfect cube factor from $x^{10}$, we again look for the highest multiple of 3 that is less than or equal to 10. This multiple is 9. We can rewrite $x^{10}$ as $x^9 \times x$. Since $x^9 = (x^3)^3$, we have a perfect cube factor of $(x^3)^3$. The remaining factor is $x$.

Simplifying $y^7$

Finally, we address the variable term $y^7$. Following the same logic, we find the highest multiple of 3 that is less than or equal to 7, which is 6. We can rewrite $y^7$ as $y^6 \times y$. Since $y^6 = (y^2)^3$, we have a perfect cube factor of $(y^2)^3$. The remaining factor is $y$.

Reassembling the Expression and Extracting Perfect Cubes

Now that we've identified the perfect cube factors and the remaining factors, we can rewrite the original expression as follows:

$\sqrt[3]{256 x^{10} y^7} = \sqrt[3]{(64 \times 4)(x^9 \times x)(y^6 \times y)}$

We can further break this down into:

$\sqrt[3]{256 x^{10} y^7} = \sqrt[3]{64 x^9 y^6 \times 4 x y}$

Using the property $\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$, we can separate the cube root:

$\sqrt[3]{256 x^{10} y^7} = \sqrt[3]{64 x^9 y^6} \times \sqrt[3]{4 x y}$

Now, we can take the cube root of the perfect cube factors:

$\sqrt[3]{64 x^9 y^6} = \sqrt[3]{4^3 (x^3)^3 (y^2)^3} = 4 x^3 y^2$

Therefore, the simplified expression becomes:

$\sqrt[3]{256 x^{10} y^7} = 4 x^3 y^2 \sqrt[3]{4 x y}$

Matching the Simplified Expression to the Options

Comparing our simplified expression, $4 x^3 y^2 \sqrt[3]{4 x y}$, to the given options, we find that it matches option B.

• A. $4 x^2 y(\sqrt[3]{x^2 y^3})$ - Incorrect
• B. $4 x^3 y^2(\sqrt[3]{4 x y})$ - Correct
• C. $16 x^3 y^2(\sqrt[3]{x y})$ - Incorrect
• D. $16 x^5 y^3(\sqrt[3]{y})$ - Incorrect

Therefore, the equivalent expression for $\sqrt[3]{256 x^{10} y^7}$ is $4 x^3 y^2(\sqrt[3]{4 x y})$. This process highlights the importance of understanding how to factor numbers and variables, identify perfect cubes, and apply the properties of radicals and exponents.

Key Takeaways and Further Exploration

In summary, simplifying expressions involving radicals often requires factoring, identifying perfect powers (in this case, perfect cubes), and applying the properties of radicals and exponents. The ability to break down complex expressions into simpler components is a crucial skill in algebra and higher-level mathematics. This particular problem involved simplifying a cube root, but the same principles can be applied to other types of radicals, such as square roots, fourth roots, and so on.

To further enhance your understanding, consider exploring additional examples of simplifying radical expressions. Practice identifying perfect squares, cubes, and other powers within radicals. You can also investigate how these simplification techniques are used in solving equations and other mathematical problems. Understanding these concepts will build a strong foundation for future mathematical endeavors. Remember, mathematics is a skill that improves with consistent practice and a willingness to explore different problem-solving strategies. This exploration not only helps you solve this specific problem but also equips you with the tools to tackle similar challenges in the future.

Additional Tips for Simplifying Radicals:

• Prime Factorization: When dealing with numerical coefficients, prime factorization can help you identify perfect power factors.
• Exponent Rules: Remember the rules of exponents, such as $a^{m+n} = a^m \times a^n$ and $(a^m)^n = a^{m*n}$, as they are essential for manipulating expressions within radicals.
• Perfect Powers: Familiarize yourself with common perfect squares, cubes, and other powers. This will help you quickly identify factors that can be extracted from radicals.
• Practice Regularly: The more you practice simplifying radicals, the more comfortable you will become with the process.

By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical problems involving radicals and exponents. Understanding these concepts will build a strong foundation for future mathematical endeavors. Remember, mathematics is a skill that improves with consistent practice and a willingness to explore different problem-solving strategies.

Conclusion

Through a careful step-by-step process, we have successfully determined that the expression equivalent to $\sqrt[3]{256 x^{10} y^7}$ is $4 x^3 y^2(\sqrt[3]{4 x y})$. This solution demonstrates the importance of understanding the properties of radicals and exponents, as well as the ability to factor expressions and identify perfect cube factors. By applying these techniques, we can simplify complex expressions and gain a deeper understanding of mathematical concepts. This detailed explanation should provide a clear understanding of the simplification process and equip you with the skills to tackle similar problems in the future. Remember to practice regularly and explore different problem-solving strategies to enhance your mathematical abilities.