Equivalent Expression For The Cube Root Of 64a⁶b⁷c⁹

Jul 14, 2025 by ADMIN 52 views

In the captivating world of mathematics, radicals often present themselves as intriguing puzzles, challenging us to unravel their hidden complexities. Among these mathematical enigmas, cube roots hold a special allure, demanding a keen understanding of their properties and nuances. This article embarks on an expedition into the realm of radicals, specifically focusing on the expression $\sqrt[3]{64 a^6 b^7 c^9}$ . Our quest is to identify the equivalent expression from a set of compelling options, namely:

A. $2 a b c^2(\sqrt[3]{4 a^2 b^3 c})$ B. $4 a^2 b^2 c^3(\sqrt[3]{b})$ C. $8 a^3 b^3 c^4(\sqrt[3]{b c})$ D. $8 a^2 b^2 c^3(\sqrt[3]{b})$

Through a meticulous step-by-step analysis, we shall embark on a journey of mathematical exploration, deciphering the secrets held within this expression and ultimately pinpointing its true equivalent.

Deconstructing the Cube Root Expression: A Journey into the Heart of Radicals

Our expedition begins with a meticulous examination of the expression $\sqrt[3]{64 a^6 b^7 c^9}$ . To conquer this challenge, we must harness the power of prime factorization and the fundamental properties of radicals. Let's break down the expression into its core components:

The Coefficient: 64 can be expressed as $4^3$ or $2^6$ , which is a perfect cube, specifically $4^3$ .
The Variable a: $a^6$ can be rewritten as $(a^2)^3$ , revealing another perfect cube.
The Variable b: $b^7$ can be expressed as $b^6 * b$ , where $b^6$ is a perfect cube, namely $(b^2)^3$ .
The Variable c: $c^9$ gracefully transforms into $(c^3)^3$ , yet another perfect cube.

With this knowledge in hand, we can rewrite the expression as follows:

$\sqrt[3]{64 a^6 b^7 c^9} = \sqrt[3]{4^3 * (a^2)^3 * b^6 * b * (c^3)^3}$

Now, let's invoke the sacred property of radicals that allows us to separate the cube root of a product into the product of cube roots:

$\sqrt[3]{4^3 * (a^2)^3 * b^6 * b * (c^3)^3} = \sqrt[3]{4^3} * \sqrt[3]{(a^2)^3} * \sqrt[3]{b^6} * \sqrt[3]{b} * \sqrt[3]{(c^3)^3}$

Each perfect cube gracefully emerges from the radical:

$4 * a^2 * b^2 * c^3 * \sqrt[3]{b}$

Thus, we arrive at the simplified expression: $4 a^2 b^2 c^3(\sqrt[3]{b})$ .

The Quest for Equivalence: Comparing Expressions and Unveiling the Truth

With our simplified expression in hand, we now embark on the quest for equivalence, comparing our result to the given options:

A. $2 a b c^2(\sqrt[3]{4 a^2 b^3 c})$ B. $4 a^2 b^2 c^3(\sqrt[3]{b})$ C. $8 a^3 b^3 c^4(\sqrt[3]{b c})$ D. $8 a^2 b^2 c^3(\sqrt[3]{b})$

A meticulous comparison reveals that option B, $4 a^2 b^2 c^3(\sqrt[3]{b})$ , perfectly matches our simplified expression. Therefore, option B is the unequivocal equivalent of the original expression.

The Final Verdict: Option B Emerges as the True Equivalent

Through our meticulous journey of mathematical exploration, we have successfully unraveled the equivalent expression for $\sqrt[3]{64 a^6 b^7 c^9}$ . By leveraging the power of prime factorization and the fundamental properties of radicals, we meticulously simplified the expression and compared it to the given options.

Our analysis has irrefutably demonstrated that option B, $4 a^2 b^2 c^3(\sqrt[3]{b})$ , stands as the true equivalent. This triumph underscores the significance of a solid understanding of radical operations and the ability to dissect complex expressions into their fundamental components.

To truly master the art of simplifying radical expressions, a deeper understanding of cube roots is essential. Cube roots, denoted by the symbol $\sqrt[3]{}$ , are the inverse operation of cubing a number. In essence, the cube root of a number 'x' is the value that, when multiplied by itself three times, equals 'x'. For instance, the cube root of 8 is 2 because 2 * 2 * 2 = 8.

Unveiling the Properties of Cube Roots

Cube roots, like their square root counterparts, possess a unique set of properties that govern their behavior. These properties serve as invaluable tools in simplifying expressions and solving equations involving cube roots.

Product Property: The cube root of a product is equal to the product of the cube roots. Mathematically, this can be expressed as:

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

This property is instrumental in breaking down complex expressions into simpler components.
Quotient Property: The cube root of a quotient is equal to the quotient of the cube roots:

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

This property allows us to simplify expressions involving fractions within cube roots.
Cube Root of a Power: The cube root of a number raised to a power can be simplified as follows:

$\sqrt[3]{a^n} = a^{\frac{n}{3}}$

This property is crucial for handling variables raised to exponents within cube roots.

The Art of Simplifying Cube Roots: A Step-by-Step Approach

Simplifying cube roots involves a systematic approach that combines prime factorization and the properties of radicals. Here's a step-by-step guide to mastering this skill:

Prime Factorization: Begin by expressing the radicand (the number or expression under the cube root) as a product of its prime factors. This decomposition reveals the building blocks of the expression.
Identify Perfect Cubes: Look for factors that appear three times within the prime factorization. These factors constitute perfect cubes.
Extract Perfect Cubes: Apply the product property of cube roots to separate the perfect cubes from the remaining factors. The cube root of a perfect cube is simply the factor itself.
Simplify the Expression: Rewrite the expression by extracting the cube roots of the perfect cubes and leaving the remaining factors under the cube root symbol.

Real-World Applications of Cube Roots: Beyond the Realm of Mathematics

Cube roots transcend the realm of abstract mathematics, finding practical applications in diverse fields:

Geometry: Cube roots play a vital role in calculating the side length of a cube given its volume. The formula $s = \sqrt[3]{V}$ relates the side length (s) to the volume (V) of a cube.
Engineering: Cube roots are used in various engineering calculations, such as determining the dimensions of spherical objects or analyzing the stress and strain in materials.
Physics: Cube roots appear in physical formulas related to volume, density, and other physical quantities.
Computer Graphics: Cube roots are employed in computer graphics to perform scaling and transformations of three-dimensional objects.

Simplifying radical expressions, especially cube roots, requires a blend of understanding the underlying principles and practicing strategic techniques. To aid in your journey toward radical mastery, here are some invaluable tips, tricks, and common pitfalls to avoid.

Essential Tips and Tricks for Radical Simplification

Master Prime Factorization: Prime factorization is the cornerstone of simplifying radicals. A solid grasp of this technique enables you to break down complex expressions into their fundamental components, revealing hidden perfect cubes or squares.
Embrace the Properties of Radicals: The product, quotient, and power properties of radicals are your allies in simplification. Utilize these properties to separate and manipulate radical expressions, paving the way for elegant solutions.
Look for Perfect Powers: Identifying perfect squares within square roots and perfect cubes within cube roots is crucial. These perfect powers can be extracted from the radical, simplifying the expression significantly.
Rationalize the Denominator: When a radical appears in the denominator of a fraction, rationalize it by multiplying both the numerator and denominator by a suitable expression that eliminates the radical from the denominator.
Practice Makes Perfect: Consistent practice is the key to mastering radical simplification. The more you work through problems, the more comfortable and confident you will become in your abilities.

Common Pitfalls to Avoid in Radical Simplification

Forgetting to Simplify: Always ensure that your final answer is in its simplest form. This means extracting all perfect powers and rationalizing the denominator, if necessary.
Incorrectly Applying Properties: Be mindful of the properties of radicals and apply them correctly. A common mistake is to incorrectly distribute a radical over a sum or difference.
Ignoring the Index: The index of the radical (the small number above the radical symbol) is crucial. A square root (index 2) requires pairs of factors, while a cube root (index 3) requires triplets.
Missing Negative Signs: When dealing with odd-indexed radicals (cube roots, fifth roots, etc.), remember that the radical of a negative number is also negative.
Rushing the Process: Simplifying radicals requires careful attention to detail. Avoid rushing through the steps, as this can lead to errors.

Our comprehensive exploration into the realm of radicals has equipped us with the knowledge and skills to confidently tackle the challenges they present. From deciphering the equivalent expression for $\sqrt[3]{64 a^6 b^7 c^9}$ to delving into the properties, simplification techniques, and real-world applications of cube roots, we have broadened our mathematical horizons.

By mastering the art of prime factorization, embracing the properties of radicals, and diligently practicing simplification techniques, we have unlocked the potential to navigate the intricate world of radicals with finesse. Remember, consistent effort and a thirst for understanding are the cornerstones of mathematical mastery. As you continue your mathematical journey, may the knowledge and insights gained here serve as a guiding light, illuminating the path to your mathematical aspirations.