Simplify: 7(cube Root Of 2x) - 3(cube Root Of 16x) - 3(cube Root Of 8x)

Hey guys! Today, we're going to tackle a fascinating problem from the realm of mathematics – simplifying radical expressions. Specifically, we'll be dissecting the expression:

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

This might look a bit intimidating at first, but don't worry! We'll break it down step by step, using a conversational approach to make sure everyone understands the underlying concepts. Think of it like we're detectives, piecing together clues to solve a mathematical mystery. Our main goal here is not just to arrive at the correct answer, but also to grasp the why behind each step. This way, you'll be equipped to handle similar problems with confidence.

Understanding the Fundamentals: Radicals and Simplification

Before we dive into the nitty-gritty, let's quickly refresh our understanding of radicals and what it means to simplify them. A radical, in its simplest form, is a mathematical expression that involves a root, such as a square root, cube root, or any nth root. The expression inside the root symbol is called the radicand. Simplifying a radical means expressing it in its most basic form, where the radicand has no perfect square factors (for square roots), perfect cube factors (for cube roots), and so on. In essence, we're trying to extract any factors from under the radical sign that we can.

When we talk about simplifying radical expressions, the key is to identify and extract perfect nth powers from the radicand. This involves factoring the radicand and looking for factors that can be written as a number raised to the power of the index of the radical. For instance, when dealing with cube roots, we're on the lookout for factors that are perfect cubes (like 8, 27, 64, etc.). Once we identify these perfect powers, we can take their nth root and bring them outside the radical sign, thus simplifying the expression.

Why is simplification important, you ask? Well, simplified expressions are easier to work with, especially when performing operations like addition, subtraction, multiplication, or division. Think of it like decluttering your workspace – a clean and organized expression allows us to see the relationships and patterns more clearly, making further calculations much smoother. Moreover, simplified forms are often preferred in mathematical conventions and are crucial for comparing and combining expressions effectively. So, mastering radical simplification isn't just about getting the right answer; it's about developing a deeper understanding of mathematical structure and efficiency.

Dissecting the Expression: A Step-by-Step Approach

Okay, let's get back to our original expression:

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

Our first step is to focus on the radicals individually and see if we can simplify them. Remember, we're looking for perfect cube factors within the radicands. Let's start with the first term, $7(\sqrt[3]{2 x})$.

• Term 1: $7(\sqrt[3]{2 x})$

  The radicand here is $2x$. Are there any perfect cube factors in $2x$? Nope! Both 2 and $x$ are already in their simplest forms, so this term is as simplified as it gets for now. We'll keep it as is and move on to the next term.

• Term 2: $-3(\sqrt[3]{16 x})$

  Now we have a radicand of $16x$. Here's where things get interesting! We need to see if 16 has any perfect cube factors. Remember, perfect cubes are numbers like 1, 8, 27, 64, and so on. Hmmm... 8 is a perfect cube ($2^3 = 8$), and it's a factor of 16! We can rewrite 16 as $8 \times 2$. So, we can rewrite the term as:

  $$-3(\sqrt[3]{8 \times 2 x})$$

  Now we can use the property of radicals that states $\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}$ to separate the cube root:

  $$-3(\sqrt[3]{8} \times \sqrt[3]{2 x})$$

  We know that $\sqrt[3]{8} = 2$, so we can simplify further:

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

  Which gives us:

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

  See how we pulled out the perfect cube factor (8) from under the radical? That's the essence of simplification!

• Term 3: $-3(\sqrt[3]{8 x})$

  Our last term has a radicand of $8x$. This one is relatively straightforward because we already know that 8 is a perfect cube! We can rewrite the term as:

  $$-3(\sqrt[3]{8} \times \sqrt[3]{x})$$

  Since $\sqrt[3]{8} = 2$, we get:

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

  Which simplifies to:

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

  Great! We've successfully simplified each of the individual radical terms.

Combining Like Terms: The Final Simplification

Now that we've simplified each term individually, let's put everything back together and see if we can combine any like terms. Remember, like terms are terms that have the same radical part. Our expression now looks like this:

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

Notice that the first two terms, $7(\sqrt[3]{2 x})$ and $-6(\sqrt[3]{2 x})$, both have the same radical part: $\sqrt[3]{2 x}$. This means we can combine them just like we would combine $7a$ and $-6a$! We simply add (or subtract) the coefficients:

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

So, our expression now becomes:

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

Can we simplify further? Nope! The radical parts, $\sqrt[3]{2 x}$ and $\sqrt[3]{x}$, are different, so we cannot combine these terms. We've reached the final simplified form.

The Grand Finale: Our Simplified Expression

After our step-by-step journey through radical simplification, we've arrived at the simplified form of the expression:

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

which is:

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

Isn't that satisfying? We took a complex-looking expression and, through careful simplification, reduced it to its most basic form. Remember, the key is to break down the problem into smaller, manageable steps, identify perfect powers within the radicals, and combine like terms. With practice, you'll become a master of radical simplification!

Key Takeaways and Tips for Success

Before we wrap up, let's highlight some crucial takeaways and tips to help you conquer radical simplification:

• Master the Perfect Powers: Knowing your perfect squares, perfect cubes, and so on is essential for quickly identifying factors within radicands. Keep a list handy as you practice!
• Factor, Factor, Factor: Breaking down the radicand into its prime factors is often the key to spotting perfect powers. Don't be afraid to write out the prime factorization.
• Use the Properties of Radicals: Remember that $\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}$ and $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$. These properties are your friends!
• Combine Like Terms: Only terms with the same radical part can be combined. Treat the radical part like a variable when combining terms.
• Practice Makes Perfect: The more you practice simplifying radical expressions, the more comfortable and confident you'll become. Don't get discouraged if you stumble at first; keep at it!

Simplifying radical expressions is a fundamental skill in mathematics, and it's a stepping stone to more advanced concepts. By understanding the underlying principles and practicing diligently, you can master this skill and unlock a whole new world of mathematical possibilities. So, keep exploring, keep questioning, and most importantly, keep having fun with math! You got this, guys!