Simplifying Radical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem! We're gonna simplify a product involving radicals. The expression we're tackling is: $\left(\sqrt{10 x^4}-x \sqrt{5 x^2}\right)\left(2 \sqrt{15 x^4}+\sqrt{3 x^3}\right)$. And, just to keep things legit, we're working under the assumption that $x \geq 0$. No negative square roots allowed here! This problem is a great way to practice simplifying radical expressions. It combines our knowledge of exponents, square roots, and basic algebra. So, grab your pencils and let's get started. By the end of this, you will have a solid understanding of how to simplify this type of expression and also gain a better grasp of the rules of working with radicals. Remember, practice is key, and the more problems you solve, the more comfortable you'll become. So, let's break this down step-by-step to make sure we don't miss any of the important details. This article will help you get familiar with the common steps you need to solve this kind of math problem. We'll be using some handy rules for radicals to make our lives easier, so let's get started!

Step 1: Simplify the Radicals

First things first, we need to simplify each radical individually. Let's start with the first term, $\sqrt{10x^4}$. We can rewrite this as $\sqrt{10} \cdot \sqrt{x^4}$. Since $x^4$ is a perfect square (because $x \geq 0$), we have $\sqrt{x^4} = x^2$. So, $\sqrt{10x^4} = x^2\sqrt{10}$. Next, let's look at the second term, $-x\sqrt{5x^2}$. Again, we can simplify $\sqrt{x^2}$ to $x$, which means $-x\sqrt{5x^2} = -x \cdot x \cdot \sqrt{5} = -x^2\sqrt{5}$. Cool, right? We've simplified the first part of our original expression.

Now, let's move on to the second part. We have $2\sqrt{15x^4} + \sqrt{3x^3}$. Similar to before, $2\sqrt{15x^4} = 2\sqrt{15} \cdot \sqrt{x^4} = 2x^2\sqrt{15}$. And then, $\sqrt{3x^3} = \sqrt{3} \cdot \sqrt{x^2} \cdot \sqrt{x} = x\sqrt{3x}$. So, by using the rule that $\sqrt{a^2} = a$ when $a \geq 0$, we have simplified all the radicals. This means our original expression is now: $(x^2\sqrt{10} - x^2\sqrt{5})(2x^2\sqrt{15} + x\sqrt{3x})$. Don't worry if it looks a bit messy; we're making progress. We've simplified all the radical expressions, and the next step is where the real fun begins: expansion!

Step 2: Expand the Expression

Alright, it's time to expand this bad boy! We're going to multiply each term in the first set of parentheses by each term in the second set. It's like distributing, but with a few extra steps. First, we multiply $x^2\sqrt{10}$ by $2x^2\sqrt{15}$. This gives us $2x^2 \cdot x^2 \cdot \sqrt{10} \cdot \sqrt{15}$. Remember that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, so $\sqrt{10} \cdot \sqrt{15} = \sqrt{150}$. Simplifying the $x$ terms, we get $2x^4\sqrt{150}$. We can further simplify $\sqrt{150}$ to $\sqrt{25 \cdot 6} = 5\sqrt{6}$. Hence, $2x^4\sqrt{150} = 2x^4 \cdot 5\sqrt{6} = 10x^4\sqrt{6}$. Cool. Next, let's multiply $x^2\sqrt{10}$ by $x\sqrt{3x}$. This gives us $x^2 \cdot x \cdot \sqrt{10} \cdot \sqrt{3x} = x^3\sqrt{30x}$. So far, we have: $10x^4\sqrt{6} + x^3\sqrt{30x}$.

Now, we move on to the second term in the first parentheses, $-x^2\sqrt{5}$. We multiply it by $2x^2\sqrt{15}$, which gives us $-x^2 \cdot 2x^2 \cdot \sqrt{5} \cdot \sqrt{15} = -2x^4\sqrt{75}$. We know that $\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}$, so $-2x^4\sqrt{75} = -2x^4 \cdot 5\sqrt{3} = -10x^4\sqrt{3}$. Finally, we multiply $-x^2\sqrt{5}$ by $x\sqrt{3x}$, which gives us $-x^2 \cdot x \cdot \sqrt{5} \cdot \sqrt{3x} = -x^3\sqrt{15x}$.

Putting it all together, our expanded expression is $10x^4\sqrt{6} + x^3\sqrt{30x} - 10x^4\sqrt{3} - x^3\sqrt{15x}$. And that, my friends, is our answer. The expansion is complete, and we've successfully simplified the expression! That's a lot of steps, but it's totally manageable once you get the hang of it. We have now expanded the original expression, using the distributive property. Now, let's check to make sure the answer is in the correct form. In the next section, we'll see if we can simplify this expression any further. Let's see how well we've done by comparing our answer with the multiple-choice options.

Step 3: Compare with the Options and Final Answer

Now, let's take a look at the multiple-choice options provided and see which one matches our simplified expression. Our final, simplified expression from the previous step is: $10x^4\sqrt{6} + x^3\sqrt{30x} - 10x^4\sqrt{3} - x^3\sqrt{15x}$. Now, we just need to compare this to the provided options.

• Option A: $10x^4\sqrt{6} + x^3\sqrt{30x} - 10x^4\sqrt{3} + x^2\sqrt{15x}$
• Option B: $10x^4\sqrt{6} + x^3\sqrt{30x} - 10x^4\sqrt{3} - x^3\sqrt{15x}$

After a careful look, we see that Option B perfectly matches our simplified expression! The signs and the terms are exactly the same. Therefore, the correct answer is B. Remember, it's super important to double-check your work, especially when dealing with multiple-choice questions. Take your time, compare each term carefully, and make sure you haven't missed a sign or coefficient somewhere along the way. Congrats! You did it. We've simplified the expression, expanded it, and compared our answer to the options. It's a great feeling, right?

In conclusion, we've successfully simplified the given radical expression. We started by simplifying the individual radicals, then expanded the expression, and finally matched our simplified answer to the options provided. The key takeaways from this problem are: how to simplify square roots, how to apply the distributive property, and the importance of carefulness throughout each step. Remember that simplifying radicals and expressions requires a solid understanding of exponent rules and the ability to manipulate the expressions effectively. Keep practicing, and you'll become a pro in no time! So, keep up the great work, and keep exploring the amazing world of mathematics! Now you should feel more confident when tackling similar problems in the future. Good luck, and keep on simplifying!