Hey math enthusiasts! Today, we're diving into a problem that involves simplifying radical expressions. We'll break down each step, making sure you understand the hows and whys. Let's get started! Our journey begins with this expression:
3 * (√(36x²y⁴(3xy)) - ∛(27x⁶y⁹)) / (x²y³)
Our goal? To simplify this beast into something more manageable. Buckle up, because this is where the fun begins. We'll tackle it piece by piece, ensuring clarity every step of the way.
Step 1: Understanding the Problem and Initial Simplification
Simplifying radical expressions often involves understanding the properties of radicals and exponents. The first thing we need to do is understand the expression itself. We have a combination of square roots, cube roots, and variables. The initial part involves simplifying within the radicals, which is where we will focus our energy first. This will help us get rid of the clutter and create a more understandable situation, like what we are seeing in the beginning of our calculation. It's about making the terms inside the radicals as simple as possible before trying to extract anything.
Alright, let's rewrite the initial expression, making sure that we tackle the parts within the radicals. This is our starting point, where we apply basic algebraic rules. We want to make sure our calculations are correct. Within the square root, we have 36x²y⁴(3xy). We'll multiply the terms inside the square root to simplify it, ensuring that we apply the correct rules of exponents.
Now, we will simplify this part: 36x²y⁴(3xy). Multiplying the terms, we get 36 * 3 * x² * x * y⁴ * y = 108x³y⁵. So, the square root part becomes √(108x³y⁵). For the cube root part, we have ∛(27x⁶y⁹). The cube root of 27 is 3. Also, the cube root of x⁶ is x², and the cube root of y⁹ is y³. This results in ∛(27x⁶y⁹) = 3x²y³. Now, let's substitute these back into our original expression:
3 * (√(108x³y⁵) - 3x²y³) / (x²y³)
We are off to a great start, guys!
Step 2: Further Simplification of the Square Root
Further simplifying radical expressions involves breaking down the terms inside the square root into perfect squares and any remaining terms that cannot be simplified further. This step will make our next move easier to pull off. The simplified form of the expression often relies on being able to recognize the perfect squares. Now let's dive in. Let's break down √(108x³y⁵) further. We can rewrite 108 as 36 * 3, where 36 is a perfect square (6²). Then, we can break down x³ into x² * x, and y⁵ into y⁴ * y, where x² and y⁴ are perfect squares. Now we have:
√(108x³y⁵) = √(36 * 3 * x² * x * y⁴ * y)
Take the perfect squares out of the square root, we get: √(36x²y⁴) * √(3xy) = 6xy²√(3xy). Now, substitute this simplified version back into the expression:
3 * (6xy²√(3xy) - 3x²y³) / (x²y³)
We are doing fantastic, aren't we?
Step 3: Division and Final Simplification
Division and final simplification are crucial parts to wrap up the problem successfully. Now, we're at the crucial stage. We must divide each term in the numerator by the denominator, x²y³. Let's go term by term. Firstly, we have 6xy²√(3xy) / x²y³. We can simplify this. The x in the numerator cancels with one x in the denominator. The y² in the numerator cancels with y² in the denominator. That gives us 6√(3xy) / xy. Next, we divide -3x²y³ by x²y³. The x²y³ in the numerator cancels completely with x²y³ in the denominator, leaving us with -3. Now, let's combine these simplified terms:
3 * (6√(3xy) / xy - 3)
Multiply each term by 3 to get the final, simplified expression:
18√(3xy) / xy - 9
And there you have it! We have successfully simplified the original radical expression. We took it apart piece by piece, made it into something simpler, and understood the logic every step of the way. That's all there is to it! That wasn't so hard, right?
Conclusion: Key Takeaways and Summary
In summary, we've seen how to simplify complex radical expressions. It's all about applying the basic rules of exponents, identifying perfect squares, and then simplifying step-by-step. We began with a seemingly difficult expression and, through breaking it down into smaller parts, managed to simplify it significantly. The key takeaways are: the importance of understanding the properties of exponents, the ability to identify perfect squares, and the practice of careful, step-by-step simplification. Always break down the problem into manageable parts and handle each part with precision. This approach will turn any complicated math problem into something that is simple and easy to solve. Keep practicing, and you will become masters of simplifying these expressions. Keep in mind that practice makes perfect, so the more you practice, the better you become. So keep at it, and don’t be scared to make mistakes. Mistakes are simply opportunities to learn and understand the concepts better. If you follow these steps, you will be acing similar problems in no time. Congratulations! You have successfully simplified the radical expression. That concludes our lesson. Remember, math is a journey of learning and discovery. Keep up the awesome work, and always have fun with it! You are all doing great!