Simplifying Cube Roots Exploring Equivalence Of Expressions For $\sqrt[3]{256 X^{10} Y^7}$

In the realm of algebra, simplifying radical expressions is a fundamental skill. This article delves into the process of simplifying a cube root expression, specifically $\sqrt[3]{256 x^{10} y^7}$. We will explore the step-by-step approach to extract perfect cubes from the radicand, ultimately arriving at the most simplified form. Understanding how to manipulate radical expressions is crucial for success in higher-level mathematics, including calculus and differential equations. This skill allows for cleaner representations of mathematical problems, which, in turn, helps solve problems in an easier and more efficient way. Let's embark on this mathematical journey together and unravel the intricacies of simplifying cube roots.

Breaking Down the Radicand: A Journey into Prime Factorization

The key to simplifying cube roots lies in identifying perfect cube factors within the radicand (the expression under the radical). Our radicand is $256 x^{10} y^7$. To find perfect cubes, we'll break down each component: the coefficient (256), the x term ($x^{10}$), and the y term ($y^7$).

First, let's tackle the coefficient, 256. Prime factorization is our tool here. We repeatedly divide 256 by prime numbers until we reach 1:

• 256 ÷ 2 = 128
• 128 ÷ 2 = 64
• 64 ÷ 2 = 32
• 32 ÷ 2 = 16
• 16 ÷ 2 = 8
• 8 ÷ 2 = 4
• 4 ÷ 2 = 2
• 2 ÷ 2 = 1

This reveals that 256 can be expressed as $2^8$. To find perfect cubes, we look for exponents that are multiples of 3. We can rewrite $2^8$ as $2^6 \cdot 2^2$, where $2^6$ is a perfect cube ($2^6 = (2^2)^3 = 4^3$). Thus, the coefficient 256 can be written as $4^3 \cdot 2^2$ or $64 \cdot 4$.

Next, we address the x term, $x^{10}$. Again, we seek exponents that are multiples of 3. We can rewrite $x^{10}$ as $x^9 \cdot x$, where $x^9$ is a perfect cube ($(x^3)^3$).

Finally, we consider the y term, $y^7$. Similarly, we rewrite $y^7$ as $y^6 \cdot y$, where $y^6$ is a perfect cube ($(y^2)^3$).

Now, we can rewrite the original radicand as: $256 x^{10} y^7 = 2^8 x^{10} y^7 = (2^6 \cdot 2^2) (x^9 \cdot x) (y^6 \cdot y) = 4^3 \cdot 4 \cdot (x^3)^3 \cdot x \cdot (y^2)^3 \cdot y$. This decomposition is the core of simplifying the cube root expression, as it isolates the perfect cube factors.

Extracting Perfect Cubes: Freeing Factors from the Radical

Now that we've identified the perfect cube factors, we can extract them from the cube root. Recall that $\sqrt[3]{a^3} = a$. Applying this principle, we can simplify the expression:

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

We can separate the cube root of a product into the product of cube roots: $\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$. Applying this, we get:

$\sqrt[3]{4^3} \cdot \sqrt[3]{4} \cdot \sqrt[3]{(x^3)^3} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3} \cdot \sqrt[3]{y}$

Now we can simplify the cube roots of the perfect cubes:

$4 \cdot \sqrt[3]{4} \cdot x^3 \cdot \sqrt[3]{x} \cdot y^2 \cdot \sqrt[3]{y}$

Rearranging the terms, we have:

$4 x^3 y^2 \cdot \sqrt[3]{4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y}$

Finally, we can combine the remaining cube roots using the property $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$:

$4 x^3 y^2 \sqrt[3]{4xy}$

Therefore, the simplified expression is $4 x^3 y^2 \sqrt[3]{4 x y}$. This process showcases the power of prime factorization and understanding exponent rules to simplify seemingly complex radical expressions. Mastering these techniques is essential for algebraic manipulation and problem-solving.

Matching the Simplified Form: Identifying the Correct Option

After simplifying the expression $\sqrt[3]{256 x^{10} y^7}$, we arrived at $4 x^3 y^2 \sqrt[3]{4 x y}$. Now, we need to compare this result with the given options to identify the equivalent expression.

The options provided are:

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

By direct comparison, we can see that option B, $4 x^3 y^2(\sqrt[3]{4 x y})$, perfectly matches our simplified result. The other options differ in either the coefficients, the exponents of the variables outside the radical, or the terms inside the cube root.

Option A has $x^2 y$ outside the radical, while our simplified expression has $x^3 y^2$. Option C has a coefficient of 16, while our expression has 4. Option D also has a coefficient of 16 and different exponents for x and y outside the radical.

Therefore, the correct answer is B. $4 x^3 y^2(\sqrt[3]{4 x y})$. This step highlights the importance of careful comparison to ensure the simplified expression is accurately matched with the correct option. A minor error in the simplification process can lead to selecting an incorrect answer, emphasizing the need for meticulous calculations and a thorough understanding of the rules of radicals.

Conclusion: The Power of Simplification

In this exploration, we successfully simplified the expression $\sqrt[3]{256 x^{10} y^7}$ to its equivalent form, $4 x^3 y^2 \sqrt[3]{4 x y}$. This process involved a series of crucial steps:

1. Breaking down the radicand into its prime factors, allowing us to identify perfect cube components.
2. Extracting perfect cubes from the radical, simplifying the expression.
3. Combining remaining terms under the radical to achieve the most simplified form.
4. Comparing the simplified form with the provided options to identify the correct answer.

The ability to simplify radical expressions is a cornerstone of algebraic proficiency. It not only allows for more concise and manageable representations of mathematical problems but also provides a deeper understanding of the underlying mathematical relationships. Mastering these simplification techniques is essential for success in various areas of mathematics and related fields. Furthermore, this exercise reinforces the importance of attention to detail and the systematic application of mathematical rules.

By understanding the principles of prime factorization, exponent rules, and the properties of radicals, we can confidently tackle a wide range of simplification problems. This skill empowers us to approach complex mathematical challenges with greater ease and accuracy. The journey of simplifying radicals is a testament to the elegance and power of mathematical manipulation. Therefore, consistent practice and a solid grasp of the fundamental concepts are the keys to unlocking the world of simplified expressions and excelling in mathematics.