Simplifying $7(\sqrt[3]{2 X})-3(\sqrt[3]{16 X})-3(\sqrt[3]{8 X})$ A Step-by-Step Guide

In the realm of mathematics, simplifying expressions involving radicals can sometimes appear daunting. However, with a systematic approach and a solid understanding of radical properties, even complex expressions can be tamed. This article delves into the process of simplifying the radical expression $7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x})$, providing a comprehensive, step-by-step guide that will not only lead you to the correct solution but also enhance your grasp of radical simplification techniques.

Understanding the Fundamentals of Radical Simplification

Before we embark on the simplification journey, it's crucial to lay the groundwork by understanding the fundamental principles governing radical expressions. Radicals, denoted by the symbol $\sqrt[n]{}$, represent the nth root of a number. The number 'n' is called the index, and the expression under the radical is called the radicand. For instance, in $\sqrt[3]{8}$, 3 is the index, and 8 is the radicand. The key to simplifying radical expressions lies in identifying perfect nth powers within the radicand. A perfect nth power is a number that can be expressed as the nth power of an integer. For example, 8 is a perfect cube (2^3), and 16 is not a perfect cube, but it contains a perfect cube factor (8).

The core idea behind simplifying radicals is to factor the radicand into its prime factors and then extract any perfect nth powers. This process relies on the property $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, which allows us to separate the radical of a product into the product of radicals. Mastering this property is essential for navigating the simplification process effectively. Let's dive into the specific expression we aim to simplify and apply these principles.

Breaking Down the Expression: A Step-by-Step Approach

Now, let's dissect the given expression: $7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x})$. The first step is to examine each term individually and identify any opportunities for simplification. We'll focus on simplifying the radicals first, and then we'll address the coefficients. The first term, $7(\sqrt[3]{2 x})$, contains the cube root of 2x. Since 2 is a prime number and x is a variable, there are no perfect cube factors within the radicand, so this term remains as is. The second term, $3(\sqrt[3]{16 x})$, presents an opportunity for simplification. The radicand 16x can be factored as $8 \cdot 2 \cdot x$. Notice that 8 is a perfect cube (2^3), which we can extract from the radical. The third term, $3(\sqrt[3]{8 x})$, also contains a perfect cube factor. The radicand 8x can be expressed as $2^3 \cdot x$, and 8 is a perfect cube. By identifying and extracting these perfect cube factors, we pave the way for simplifying the entire expression.

Step 1: Simplifying the Second Term $3(\sqrt[3]{16 x})$

Let's focus on simplifying the second term, $3(\sqrt[3]{16 x})$. As we discussed earlier, 16 can be factored as $8 \cdot 2$, where 8 is a perfect cube (2^3). Therefore, we can rewrite the term as follows:

$3(\sqrt[3]{16 x}) = 3(\sqrt[3]{8 \cdot 2 \cdot x})$

Now, we apply the property $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ to separate the radical:

$3(\sqrt[3]{8 \cdot 2 \cdot x}) = 3(\sqrt[3]{8} \cdot \sqrt[3]{2 x})$

Since $\sqrt[3]{8} = 2$, we can substitute this value:

$3(\sqrt[3]{8} \cdot \sqrt[3]{2 x}) = 3(2 \cdot \sqrt[3]{2 x}) = 6(\sqrt[3]{2 x})$

Thus, the simplified form of the second term is $6(\sqrt[3]{2 x})$. This simplification significantly reduces the complexity of the term and brings us closer to simplifying the entire expression.

Step 2: Simplifying the Third Term $3(\sqrt[3]{8 x})$

Next, we'll simplify the third term, $3(\sqrt[3]{8 x})$. As we identified earlier, 8 is a perfect cube (2^3). So, we can rewrite the term as:

$3(\sqrt[3]{8 x}) = 3(\sqrt[3]{2^3 \cdot x})$

Applying the property $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$, we separate the radical:

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

Since $\sqrt[3]{2^3} = 2$, we substitute this value:

$3(\sqrt[3]{2^3} \cdot \sqrt[3]{x}) = 3(2 \cdot \sqrt[3]{x}) = 6(\sqrt[3]{x})$

Therefore, the simplified form of the third term is $6(\sqrt[3]{x})$. This simplification further reduces the complexity of the expression and sets the stage for combining like terms.

Combining Like Terms: The Final Simplification

Now that we've simplified each term individually, we can rewrite the original expression with the simplified terms:

$7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x}) = 7(\sqrt[3]{2 x}) - 6(\sqrt[3]{2 x}) - 6(\sqrt[3]{x})$

Notice that the first two terms, $7(\sqrt[3]{2 x})$ and $-6(\sqrt[3]{2 x})$, are like terms because they both contain the radical $\sqrt[3]{2 x}$. We can combine these terms by subtracting their coefficients:

$7(\sqrt[3]{2 x}) - 6(\sqrt[3]{2 x}) = (7 - 6)(\sqrt[3]{2 x}) = 1(\sqrt[3]{2 x}) = \sqrt[3]{2 x}$

Now, we substitute this simplified result back into the expression:

$\sqrt[3]{2 x} - 6(\sqrt[3]{x})$

The resulting expression, $\sqrt[3]{2 x} - 6(\sqrt[3]{x})$, is the simplified form of the original expression. There are no further like terms to combine, and the radicals are in their simplest form.

The Answer and Why It Matters

Therefore, the simplified form of the expression $7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x})$ is $\sqrt[3]{2 x} - 6(\sqrt[3]{x})$, which corresponds to option C.

This exercise highlights the importance of mastering radical simplification techniques. Simplifying expressions not only makes them easier to work with but also reveals underlying relationships and structures that might otherwise remain hidden. In various fields, from physics to engineering, simplifying mathematical expressions is a crucial skill for problem-solving and analysis. By understanding the principles and applying a systematic approach, you can confidently tackle even the most complex radical expressions.

Common Pitfalls to Avoid

While simplifying radical expressions, several common pitfalls can lead to errors. One frequent mistake is failing to identify perfect nth powers within the radicand. For instance, when simplifying $\sqrt[3]{16 x}$, some might overlook the fact that 16 contains a perfect cube factor (8). Another common error is incorrectly applying the property $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Remember that this property only applies to multiplication and not to addition or subtraction. A third pitfall is incorrectly combining terms that are not like terms. Only terms with the same radical expression can be combined. To avoid these mistakes, practice is key. Work through numerous examples, paying close attention to each step, and double-check your work to ensure accuracy.

Practice Problems to Sharpen Your Skills

To solidify your understanding of simplifying radical expressions, try tackling these practice problems:

1. Simplify $5(\sqrt[3]{24 x}) + 2(\sqrt[3]{3 x})$
2. Simplify $4(\sqrt[3]{81 x}) - (\sqrt[3]{3 x})$
3. Simplify $2(\sqrt[3]{54 x}) + 3(\sqrt[3]{16 x}) - (\sqrt[3]{2 x})$

By working through these problems, you'll reinforce the concepts and techniques discussed in this article and gain confidence in your ability to simplify radical expressions. Remember to break down each term, identify perfect nth powers, extract them from the radicals, and combine like terms. With consistent practice, you'll become proficient in simplifying radical expressions and unlock a valuable skill for your mathematical toolkit.

Conclusion: Mastering the Art of Simplification

Simplifying radical expressions is an essential skill in mathematics, with applications spanning various fields. By understanding the fundamental principles, employing a systematic approach, and avoiding common pitfalls, you can confidently tackle even the most complex expressions. This article has provided a comprehensive guide to simplifying the expression $7(\sqrt[3]{2 x})-3(\sqrt[3]{16 x})-3(\sqrt[3]{8 x})$, illustrating each step with clarity and precision. Remember to practice regularly, and you'll master the art of simplification, unlocking a powerful tool for your mathematical journey. Embracing the challenge of simplifying expressions will not only enhance your problem-solving abilities but also deepen your appreciation for the elegance and structure inherent in mathematics.