Simplifying The Cube Root Sum 2(³√16x³y) + 4(³√54x⁶y⁵)

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Hey there, math enthusiasts! Today, we're going to tackle a fascinating sum involving cube roots: 2(³√16x³y) + 4(³√54x⁶y⁵). This might look a bit intimidating at first glance, but don't worry, we'll break it down step by step and unveil the solution together. Our goal is not just to find the answer, but to truly understand the underlying principles of simplifying radical expressions, especially those involving cube roots. So, buckle up and get ready for a journey into the world of algebraic manipulation!

Diving into the Realm of Cube Roots

Before we jump into the problem, let's refresh our understanding of cube roots. Remember, the cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Similarly, the cube root of 27 is 3 because 3 * 3 * 3 = 27. Now, when we encounter expressions with variables and coefficients under the cube root, things get a little more interesting. We need to look for perfect cubes within the expression to simplify it effectively. Think of perfect cubes like 1, 8, 27, 64, 125, and so on. These are numbers that have whole number cube roots. Identifying these perfect cubes is the key to simplifying our expression.

Identifying Perfect Cubes within Radicals

Now, let's focus on the expression under the cube root. We need to identify the perfect cubes lurking within 16x³y and 54x⁶y⁵. For 16, we can rewrite it as 8 * 2, where 8 is a perfect cube (2³). The term x³ is already a perfect cube. As for y, it's just y to the power of 1, so we can't extract a perfect cube from it just yet. Now, let's tackle 54. We can break 54 down into 27 * 2, where 27 is a perfect cube (3³). The term x⁶ is also a perfect cube because x⁶ = (x²)³. For y⁵, we can rewrite it as y³ * y², where y³ is a perfect cube. See how we're strategically breaking down the numbers and variables to reveal those perfect cubes? This is the core of simplifying these kinds of expressions. Once we've identified the perfect cubes, we can pull them out of the cube root, making the expression much cleaner and easier to work with.

Deconstructing the First Term: 2(³√16x³y)

Let's begin our simplification journey with the first term: 2(³√16x³y). Remember our earlier detective work? We found that 16 can be expressed as 8 * 2, where 8 is the perfect cube 2³. So, we can rewrite our term as 2(³√8 * 2 * x³ * y). Now, we can use the property of radicals that states ³√(a * b) = ³√a * ³√b. Applying this, we get 2(³√8 * ³√2 * ³√x³ * ³√y). The cube root of 8 is 2, and the cube root of x³ is x. So, we can simplify further to 2 * 2 * x * ³√2y, which simplifies to 4x(³√2y). See how we methodically broke down the expression, identified the perfect cubes, and extracted them from the radical? This is the power of understanding the properties of radicals!

Step-by-Step Breakdown

To make sure we're all on the same page, let's quickly recap the steps we took to simplify the first term:

  1. Rewrite 16 as 8 * 2: This allowed us to identify the perfect cube, 8.
  2. Apply the property of radicals: We separated the cube root of the product into the product of cube roots.
  3. Extract the cube roots: We found the cube root of 8 (which is 2) and the cube root of x³ (which is x).
  4. Simplify: We multiplied the constants together to get the final simplified form: 4x(³√2y).

This step-by-step approach is crucial for tackling complex expressions. By breaking them down into smaller, manageable chunks, we can avoid making mistakes and gain a deeper understanding of the process. Remember, math isn't about memorizing formulas; it's about understanding the logic behind them.

Taming the Second Term: 4(³√54x⁶y⁵)

Now, let's turn our attention to the second term: 4(³√54x⁶y⁵). Just like before, our first task is to identify the perfect cubes lurking within. We know that 54 can be rewritten as 27 * 2, where 27 is the perfect cube 3³. The term x⁶ can be expressed as (x²)³, which is also a perfect cube. And for y⁵, we can rewrite it as y³ * y², where y³ is a perfect cube. Armed with this knowledge, we can rewrite the term as 4(³√27 * 2 * x⁶ * y³ * y²). Now, let's apply the property of radicals and separate the cube roots: 4(³√27 * ³√2 * ³√x⁶ * ³√y³ * ³√y²). The cube root of 27 is 3, the cube root of x⁶ is x², and the cube root of y³ is y. So, we can simplify further to 4 * 3 * x² * y * ³√2y², which simplifies to 12x²y(³√2y²). We've successfully tamed the second term!

A Closer Look at the Process

Let's break down the steps we took to simplify the second term, just like we did with the first:

  1. Rewrite 54 as 27 * 2 and y⁵ as y³ * y²: This allowed us to identify the perfect cubes, 27 and y³.
  2. Express x⁶ as (x²)³: This made it clear that x⁶ is also a perfect cube.
  3. Apply the property of radicals: We separated the cube root of the product into the product of cube roots.
  4. Extract the cube roots: We found the cube root of 27 (which is 3), the cube root of x⁶ (which is x²), and the cube root of y³ (which is y).
  5. Simplify: We multiplied the constants and variables together to get the final simplified form: 12x²y(³√2y²).

Notice the pattern? By consistently applying the properties of radicals and identifying perfect cubes, we can simplify even the most complex-looking expressions. This systematic approach is the key to success in algebra and beyond.

Putting it All Together: The Final Sum

Now that we've successfully simplified both terms, it's time to add them together. We found that 2(³√16x³y) = 4x(³√2y) and 4(³√54x⁶y⁵) = 12x²y(³√2y²). So, our original sum, 2(³√16x³y) + 4(³√54x⁶y⁵), can now be written as 4x(³√2y) + 12x²y(³√2y²). And there you have it! We've successfully simplified the original expression and arrived at our final answer. This might seem like a long journey, but by breaking down the problem into smaller steps and understanding the underlying principles, we were able to conquer it together.

The Beauty of Simplification

Isn't it amazing how a seemingly complex expression can be simplified into something much more manageable? This is the beauty of mathematical simplification. It allows us to see the underlying structure and relationships within an expression, making it easier to understand and work with. In this case, by identifying perfect cubes and applying the properties of radicals, we transformed a complicated sum into a clear and concise expression. This ability to simplify is not just a mathematical skill; it's a valuable problem-solving skill that can be applied in many areas of life.

The Answer and Key Takeaways

So, to answer the question: What is the following sum? 2(³√16x³y) + 4(³√54x⁶y⁵)

The answer is 4x(³√2y) + 12x²y(³√2y²), which corresponds to option A.

Key Takeaways From Our Math Adventure

Before we wrap up, let's recap the key takeaways from our mathematical adventure today:

  • Identify Perfect Cubes: When simplifying cube roots, the first step is to identify perfect cubes within the expression under the radical.
  • Apply the Property of Radicals: Remember that ³√(a * b) = ³√a * ³√b. This property allows us to separate the cube root of a product into the product of cube roots.
  • Break it Down: Complex expressions can be overwhelming. Break them down into smaller, manageable steps to avoid mistakes and gain a deeper understanding.
  • Practice Makes Perfect: The more you practice simplifying radical expressions, the more comfortable and confident you'll become.

So, keep exploring the world of mathematics, guys, and never stop asking questions! Math is not just about numbers and equations; it's about problem-solving, critical thinking, and the joy of discovery.