Simplifying Cube Root Product A Step-by-Step Guide

Delving into the realm of mathematics, we often encounter expressions that seem complex at first glance. However, with a systematic approach and a solid understanding of fundamental principles, we can unravel these intricacies and arrive at elegant solutions. In this comprehensive exploration, we will dissect the product of cube roots, $\sqrt[3]{16 x^7} \sqrt[3]{12 x^9}$, employing a step-by-step methodology to simplify the expression and identify the correct answer from the given options. Our journey will involve leveraging the properties of radicals and exponents, ultimately revealing the underlying structure of this mathematical puzzle.

Demystifying Cube Roots and Their Properties

Before we embark on the simplification process, it is crucial to grasp the concept of cube roots and their associated properties. A cube root of a number is a value that, when multiplied by itself three times, yields the original number. For instance, the cube root of 8 is 2, as 2 * 2 * 2 = 8. Mathematically, we represent the cube root of a number 'a' as $\sqrt[3]{a}$. Understanding this fundamental definition is paramount to manipulating expressions involving cube roots.

One of the key properties of radicals, including cube roots, is the product rule. This rule states that the nth root of a product is equal to the product of the nth roots of the individual factors. In mathematical notation, this can be expressed as: $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$. This property serves as a cornerstone in simplifying expressions involving the product of radicals. By applying this rule, we can break down complex expressions into simpler, more manageable components.

Another essential property involves simplifying radicals with exponents. Specifically, for cube roots, we can extract factors that are perfect cubes from under the radical. This means that if we have a term like $\sqrt[3]{a^3}$, it simplifies to 'a'. Similarly, $\sqrt[3]{a^6}$ simplifies to $a^2$, and so on. This principle is derived from the definition of cube roots and the properties of exponents. By identifying and extracting perfect cube factors, we can significantly simplify radical expressions.

Step-by-Step Simplification of $\sqrt[3]{16 x^7} \sqrt[3]{12 x^9}$

Now, armed with a firm grasp of cube root properties, we can embark on the simplification of the given expression, $\sqrt[3]{16 x^7} \sqrt[3]{12 x^9}$. Our approach will involve a series of logical steps, each building upon the previous one, to systematically reduce the expression to its simplest form.

Step 1: Applying the Product Rule of Radicals

The first step in our simplification process is to apply the product rule of radicals. This allows us to combine the two cube roots into a single cube root containing the product of the radicands. Applying the rule $\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$, we can rewrite the expression as:

$\sqrt[3]{16 x^7} \sqrt[3]{12 x^9} = \sqrt[3]{(16 x^7)(12 x^9)}$

This step consolidates the expression, making it easier to manipulate in subsequent steps. By combining the radicands under a single cube root, we streamline the simplification process and set the stage for further analysis.

Step 2: Multiplying the Radicands

Next, we proceed to multiply the radicands within the cube root. This involves multiplying the numerical coefficients and applying the rules of exponents to the variable terms. Recall that when multiplying exponents with the same base, we add the powers. Thus, $x^7 * x^9 = x^(7+9) = x^16$. Performing the multiplication, we get:

$\sqrt[3]{(16 x^7)(12 x^9)} = \sqrt[3]{192 x^{16}}$

This step simplifies the expression further by combining like terms and reducing the number of individual factors under the cube root. We now have a single cube root containing a numerical coefficient and a variable term with an exponent.

Step 3: Factoring the Radicand

To simplify the cube root, we need to factor the radicand, 192x^16, into its prime factors and identify any perfect cubes. The prime factorization of 192 is $2^6 * 3$. For the variable term, we can express $x^16$ as $x^{15} * x$, where $x^{15}$ is a perfect cube since 15 is divisible by 3. Rewriting the expression with the factored radicand, we have:

$\sqrt[3]{192 x^{16}} = \sqrt[3]{2^6 * 3 * x^{15} * x}$

This step is crucial for identifying and extracting perfect cube factors from under the radical. By factoring the radicand, we expose the components that can be simplified, paving the way for the final simplification steps.

Step 4: Extracting Perfect Cubes

Now, we can extract the perfect cubes from under the cube root. Recall that $\sqrt[3]{a^3} = a$. We have $2^6 = (2^2)^3$, $x^{15} = (x^5)^3$. Extracting these perfect cubes, we get:

$\sqrt[3]{2^6 * 3 * x^{15} * x} = \sqrt[3]{(2^2)^3 * 3 * (x^5)^3 * x} = 2^2 * x^5 * \sqrt[3]{3x} = 4 x^5 \sqrt[3]{3 x}$

This step is the culmination of our simplification efforts. By extracting the perfect cubes, we have reduced the expression to its simplest form, with only the non-perfect cube factors remaining under the radical.

Identifying the Correct Answer

Comparing our simplified expression, $4 x^5 \sqrt[3]{3 x}$, with the given options, we can clearly see that it matches option D. Therefore, the correct answer is:

D. $4 x^5(\sqrt[3]{3 x})$

Conclusion: A Triumph of Mathematical Reasoning

Through a meticulous step-by-step simplification process, we have successfully navigated the intricacies of the expression $\sqrt[3]{16 x^7} \sqrt[3]{12 x^9}$. By applying the properties of radicals and exponents, we have transformed the seemingly complex expression into its simplest form, revealing the underlying mathematical elegance. This exercise underscores the power of systematic reasoning and the importance of a solid foundation in mathematical principles. By mastering these concepts, we can confidently tackle a wide range of mathematical challenges.

This journey through cube root simplification serves as a testament to the beauty and power of mathematics. By understanding the fundamental principles and applying them methodically, we can unlock the solutions to even the most intricate problems. The satisfaction of arriving at the correct answer through logical deduction is a reward in itself, fueling our passion for mathematical exploration and discovery.