Factoring $3((2a)^3 - (3b)^3)$ A Step-by-Step Guide

Hey guys! Today, we're diving deep into a fascinating math problem: $3((2a)^3 - (3b)^3)$. This expression looks intimidating at first glance, but don't worry, we'll break it down step by step and unlock its secrets. This isn't just about solving a problem; it's about understanding the underlying principles of factoring and how they can be applied in various mathematical scenarios. So, buckle up, grab your thinking caps, and let's get started!

Understanding the Expression: What are we dealing with?

Before we jump into solving, let's dissect the expression. $3((2a)^3 - (3b)^3)$ might seem like a jumble of numbers and letters, but it's actually a beautiful example of the difference of cubes pattern. This pattern is a fundamental concept in algebra, and mastering it will open doors to solving many similar problems. At its core, the expression represents 3 times the difference between two cubes: $(2a)^3$ and $(3b)^3$. Recognizing this structure is the first crucial step in simplifying the expression. We have a constant multiplier of 3 outside the parentheses, and inside, we have the difference between the cube of $2a$ and the cube of $3b$. The cube of a term means that the term is multiplied by itself three times. So, $(2a)^3$ is the same as $(2a) * (2a) * (2a)$, and similarly, $(3b)^3$ is $(3b) * (3b) * (3b)$. Understanding this notation is key to proceeding with the simplification. The difference of cubes is a special algebraic pattern that allows us to factor expressions in a specific way. This pattern is particularly useful because it provides a direct method for breaking down complex expressions into simpler ones, making them easier to work with. In this case, recognizing the difference of cubes pattern allows us to apply a known formula that will help us factor the expression and ultimately simplify it. So, the key takeaway here is that we're dealing with a difference of cubes expression, and we'll use a specific factoring formula to tackle it. By understanding this structure, we can approach the problem strategically and efficiently.

The Magic Formula: Difference of Cubes Unveiled

Now, for the star of the show: the difference of cubes formula! This formula is our secret weapon for tackling expressions like the one we have. The formula states that: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. This might look a bit intimidating at first, but trust me, it's simpler than it seems. This formula provides a direct way to factor expressions that fit the difference of cubes pattern. It essentially breaks down the difference of two cubes into the product of a binomial $(a - b)$ and a trinomial $(a^2 + ab + b^2)$. The beauty of this formula lies in its ability to transform a complex expression into a more manageable form, making it easier to simplify or solve for unknowns. To effectively use this formula, we need to correctly identify the 'a' and 'b' terms in our expression. These terms are the cube roots of the two terms in the original difference of cubes expression. Once we've identified 'a' and 'b', we can simply plug them into the formula and expand the resulting expression. This process will give us the factored form of the original expression, which can then be further simplified if needed. Understanding and applying the difference of cubes formula is a fundamental skill in algebra, and it will prove invaluable in solving various mathematical problems. By mastering this formula, you'll be able to confidently tackle expressions that involve the difference of cubes and simplify them with ease. So, remember this formula – it's your key to unlocking the secrets of expressions like $3((2a)^3 - (3b)^3)$. In our case, 'a' corresponds to $2a$ and 'b' corresponds to $3b$. We'll use this knowledge to apply the formula in the next step. This formula is a cornerstone of factoring, so let's keep it in our toolbox!

Applying the Formula: Step-by-Step Breakdown

Okay, let's get our hands dirty and apply the formula to our expression. Remember, we have $3((2a)^3 - (3b)^3)$ and our formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. First, we need to identify what 'a' and 'b' are in our specific case. As we discussed earlier, 'a' is $2a$ and 'b' is $3b$. Now, we simply substitute these values into the formula. This gives us: $(2a)^3 - (3b)^3 = (2a - 3b)((2a)^2 + (2a)(3b) + (3b)^2)$. See? We're already making progress! We've successfully applied the difference of cubes formula and transformed our expression into a factored form. This step is crucial because it breaks down the complex difference of cubes into a product of simpler terms, making it easier to work with. Now, we need to simplify the terms inside the parentheses. Let's start with the first set of parentheses: $(2a - 3b)$. This term is already in its simplest form, so we can leave it as is. Next, let's tackle the second set of parentheses: $((2a)^2 + (2a)(3b) + (3b)^2)$. We need to expand and simplify each term within this expression. $(2a)^2$ is equal to $4a^2$, $(2a)(3b)$ is equal to $6ab$, and $(3b)^2$ is equal to $9b^2$. So, the expression inside the second set of parentheses simplifies to: $4a^2 + 6ab + 9b^2$. Now, we have the fully factored form of the expression inside the original parentheses. But don't forget, we still have that 3 outside! So, the next step is to multiply the entire expression by 3 to get our final simplified form. This will give us the complete solution to our problem.

Simplifying Further: Let's Clean Things Up

We're almost there! We have $3((2a - 3b)(4a^2 + 6ab + 9b^2))$. Now, we just need to distribute that 3 across the factored expression. This means multiplying the entire expression inside the parentheses by 3. We can choose to multiply the 3 by either the first factor $(2a - 3b)$ or the second factor $(4a^2 + 6ab + 9b^2)$. It doesn't matter which one we choose, as the final result will be the same. For simplicity, let's multiply the 3 by the first factor, $(2a - 3b)$. This gives us: $3(2a - 3b) = 6a - 9b$. Now, we have: $(6a - 9b)(4a^2 + 6ab + 9b^2)$. This is our final factored form of the expression. We've successfully applied the difference of cubes formula, simplified the expression, and arrived at our solution. You might be tempted to multiply these two factors together, but that would actually take us back to the original expression. The goal of factoring is to break down an expression into its simplest components, so we want to leave it in this factored form. This form allows us to see the underlying structure of the expression and can be useful for solving equations or simplifying other expressions that involve this factor. So, the key takeaway here is that we've successfully factored the expression and arrived at our final answer: $(6a - 9b)(4a^2 + 6ab + 9b^2)$. This demonstrates the power of the difference of cubes formula and how it can be used to simplify complex algebraic expressions.

The Final Answer: Ta-da!

So, after all that work, what's our final answer? Drumroll, please... It's $(6a - 9b)(4a^2 + 6ab + 9b^2)$! We did it, guys! We successfully factored the expression $3((2a)^3 - (3b)^3)$ using the difference of cubes formula. This wasn't just about getting the right answer; it was about understanding the process and the underlying mathematical principles. We started by recognizing the structure of the expression as a difference of cubes. Then, we unleashed our secret weapon: the difference of cubes formula. We carefully applied the formula, step by step, substituting the correct values and simplifying the resulting expression. Finally, we arrived at our factored form, which represents the original expression in a more simplified and insightful way. This process highlights the importance of pattern recognition in mathematics. By recognizing the difference of cubes pattern, we were able to apply a specific formula that made the problem much easier to solve. This skill is crucial for tackling more complex mathematical problems in the future. So, remember the difference of cubes formula and practice applying it to various expressions. The more you practice, the more comfortable you'll become with the formula and the faster you'll be able to solve these types of problems. And most importantly, remember that math isn't just about memorizing formulas; it's about understanding the logic and reasoning behind them. So, keep exploring, keep questioning, and keep unlocking the mysteries of mathematics!