Simplifying Radicals: Find The Quotient Of The Expression

Have you ever stumbled upon a mathematical expression that looks intimidating at first glance? Expressions involving radicals, especially cube roots, can sometimes seem like a puzzle. But don't worry, we're here to break it down step by step. In this article, we'll tackle the expression $\frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9}}$ and find its simplified quotient. Get ready to sharpen your math skills and discover the beauty of simplifying radicals!

Understanding the Problem

Before we dive into the solution, let's make sure we understand what the question is asking. We have a fraction where the numerator is $6 - 3(\sqrt[3]{6})$ and the denominator is $\sqrt[3]{9}$. Our goal is to simplify this expression and find an equivalent form that matches one of the given options. This involves dealing with cube roots and rationalizing the denominator, which might sound complex, but we'll take it one step at a time.

Key Concepts: Radicals and Cube Roots

First, let's clarify what we mean by radicals and cube roots. A radical is a mathematical expression that involves a root, such as a square root, cube root, or any nth root. The cube root of a number x is a value that, when multiplied by itself three times, equals x. For example, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8$. Understanding these basics is crucial for simplifying our expression.

Why Simplify?

Simplifying radical expressions makes them easier to work with and compare. In many cases, it also helps us identify patterns and relationships that might not be obvious in the original form. In our specific problem, simplifying will help us match the expression to one of the provided answer choices, which are all in a simplified radical form. Simplifying radicals is a fundamental skill in algebra and is used extensively in various mathematical fields.

The Challenge: Rationalizing the Denominator

One of the main challenges in this problem is the presence of a radical in the denominator. In mathematics, it's generally preferred to have a rational denominator (i.e., a denominator without any radicals). The process of removing radicals from the denominator is called rationalizing the denominator. We'll use this technique to simplify our expression and make it easier to handle. Rationalizing the denominator often involves multiplying the numerator and denominator by a suitable radical expression that will eliminate the radical in the denominator. This process is crucial for simplifying expressions and making them easier to compare and manipulate.

Step-by-Step Solution

Now, let's walk through the solution step by step. We'll break down each step and explain the reasoning behind it.

Step 1: Rationalizing the Denominator

The first step is to rationalize the denominator. Our denominator is $\sqrt[3]{9}$. To eliminate the cube root, we need to multiply the denominator by something that will result in a perfect cube. Since $9 = 3^2$, we can multiply by $\sqrt[3]{3}$ to get $\sqrt[3]{9} \times \sqrt[3]{3} = \sqrt[3]{27} = 3$, which is a rational number.

To keep the fraction equivalent, we must also multiply the numerator by the same term, $\sqrt[3]{3}$. So, we have:

$$\frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9}} \times \frac{\sqrt[3]{3}}{\sqrt[3]{3}}$$

This step is crucial because it eliminates the radical from the denominator, making the expression easier to simplify further. Multiplying both the numerator and denominator by the same term ensures that we are not changing the value of the original expression, only its form.

Step 2: Distribute in the Numerator

Next, we distribute $\sqrt[3]{3}$ in the numerator:

$$\frac{(6-3(\sqrt[3]{6}))(\sqrt[3]{3})}{\sqrt[3]{9} \times \sqrt[3]{3}} = \frac{6(\sqrt[3]{3}) - 3(\sqrt[3]{6})(\sqrt[3]{3})}{3}$$

Here, we've multiplied $\sqrt[3]{3}$ by both terms in the numerator. This is a straightforward application of the distributive property. It's important to be careful with the multiplication of radicals; we'll simplify the term $3(\sqrt[3]{6})(\sqrt[3]{3})$ in the next step.

Step 3: Simplify the Radicals

Now, let's simplify the radicals in the numerator. We have:

$$3(\sqrt[3]{6})(\sqrt[3]{3}) = 3(\sqrt[3]{6 \times 3}) = 3(\sqrt[3]{18})$$

So, the numerator becomes:

$$6(\sqrt[3]{3}) - 3(\sqrt[3]{18})$$

In this step, we used the property that $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$. This property allows us to combine the radicals into a single radical, which can then be simplified if possible. Here, we combined $\sqrt[3]{6}$ and $\sqrt[3]{3}$ to get $\sqrt[3]{18}$.

Step 4: Divide by the Denominator

We now have the expression:

$$\frac{6(\sqrt[3]{3}) - 3(\sqrt[3]{18})}{3}$$

We can divide each term in the numerator by the denominator, 3:

$$\frac{6(\sqrt[3]{3})}{3} - \frac{3(\sqrt[3]{18})}{3} = 2(\sqrt[3]{3}) - \sqrt[3]{18}$$

This step involves simplifying the fraction by dividing each term in the numerator by the common denominator. This is a standard algebraic simplification technique. Dividing both terms by 3 makes the expression much cleaner and easier to compare with the answer choices.

Step 5: Match the Answer

Our simplified expression is $2(\sqrt[3]{3}) - \sqrt[3]{18}$, which matches option A.

Final Answer: A. $2(\sqrt[3]{3}) - \sqrt[3]{18}$

Common Mistakes to Avoid

When simplifying radical expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Incorrectly Rationalizing the Denominator: Make sure you multiply by the correct radical to eliminate the root in the denominator. For cube roots, you need to make the expression under the root a perfect cube.
• Improper Distribution: When multiplying a radical expression by a sum or difference, ensure you distribute correctly to each term.
• Forgetting to Simplify Radicals: Always check if the radicals can be simplified further. Look for perfect cube factors within the radical.
• Arithmetic Errors: Simple arithmetic mistakes can throw off your entire solution. Double-check your calculations at each step.
• Not Matching the Answer Format: Sometimes, you might simplify the expression correctly but not be able to match it to the given options. This often means you need to simplify further or rewrite the expression in a different form.

Practice Problems

To solidify your understanding, let's look at a couple of practice problems.

Practice Problem 1: Simplify the expression $\frac{10}{2(\sqrt[3]{2})}$.

Solution:

1. Rationalize the denominator by multiplying by $\frac{\sqrt[3]{4}}{\sqrt[3]{4}}$:

   $$\frac{10}{2(\sqrt[3]{2})} \times \frac{\sqrt[3]{4}}{\sqrt[3]{4}} = \frac{10(\sqrt[3]{4})}{2(\sqrt[3]{8})}$$

2. Simplify:

   $$\frac{10(\sqrt[3]{4})}{2 \times 2} = \frac{10(\sqrt[3]{4})}{4} = \frac{5(\sqrt[3]{4})}{2}$$

Practice Problem 2: Simplify the expression $\frac{5 + \sqrt[3]{3}}{\sqrt[3]{9}}$.

Solution:

1. Rationalize the denominator by multiplying by $\frac{\sqrt[3]{3}}{\sqrt[3]{3}}$:

   $$\frac{5 + \sqrt[3]{3}}{\sqrt[3]{9}} \times \frac{\sqrt[3]{3}}{\sqrt[3]{3}} = \frac{(5 + \sqrt[3]{3})(\sqrt[3]{3})}{\sqrt[3]{27}}$$

2. Distribute in the numerator:

   $$\frac{5(\sqrt[3]{3}) + \sqrt[3]{9}}{3}$$

These practice problems should give you a good sense of how to approach similar questions. Remember, the key is to break down the problem into smaller, manageable steps.

Conclusion

Simplifying radical expressions, like the one we tackled today, might seem daunting at first. However, by breaking the problem down into manageable steps and understanding the underlying concepts, we can confidently find the solution. Remember the key techniques: rationalizing the denominator, distributing terms, and simplifying radicals. With practice, you'll become more comfortable and proficient in handling these types of problems. Keep practicing, and you'll find that even the most complex expressions can be simplified with the right approach!

For further learning and practice on simplifying radicals, you can visit Khan Academy's Radical Expressions Section.