Multiplying Expressions With Square Roots: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of multiplying expressions that involve square roots. Specifically, we're going to break down how to multiply $(5 \sqrt{x}+6)(5 \sqrt{x}+5)$, assuming that all variables represent positive real numbers. This might seem a little intimidating at first, but don't worry! We'll take it step by step, and by the end, you'll be a pro at this. So, grab your pencils, and let's get started!

Understanding the Basics

Before we jump into the problem, let's quickly review some fundamental concepts. When we're dealing with expressions like $(5 \sqrt{x}+6)(5 \sqrt{x}+5)$, we need to remember the distributive property, often remembered by the acronym FOIL which stands for First, Outer, Inner, Last. This property helps us multiply two binomials (expressions with two terms) correctly. In simpler terms, it means that each term in the first set of parentheses needs to be multiplied by each term in the second set of parentheses. This ensures that we account for all possible combinations and simplify the expression accurately.

Also, it's crucial to remember how square roots work. A square root of a number is a value that, when multiplied by itself, gives you the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9. When multiplying square roots, like $\sqrt{x} * \sqrt{x}$, we get x, assuming x is positive. This is because $\sqrt{x}$ multiplied by itself is simply x. This understanding of square roots is vital for simplifying our expressions after we've applied the distributive property. Without it, we'd be stuck with unsimplified terms, and nobody wants that!

Keeping these basic principles in mind, we're well-equipped to tackle the multiplication of our expression. It's like having the right tools for the job – with the distributive property and our knowledge of square roots, we can confidently approach this problem and break it down into manageable steps. Let's move on to the next section where we'll actually start the multiplication process. Remember, the key is to take it one step at a time and not rush through. We've got this!

Applying the FOIL Method

Alright, let's get our hands dirty and apply the FOIL method to our expression: $(5 \sqrt{x}+6)(5 \sqrt{x}+5)$. Remember, FOIL stands for First, Outer, Inner, Last, which tells us the order in which we need to multiply the terms. It's like a little roadmap for multiplying binomials, ensuring we don't miss any combinations.

• First: Multiply the first terms in each parenthesis. That's $(5 \sqrt{x}) * (5 \sqrt{x})$. When we multiply these, we get $25x$. Remember that $\sqrt{x} * \sqrt{x}$ equals x, and 5 times 5 is 25. So, the first part is $25x$.
• Outer: Next, we multiply the outer terms, which are $(5 \sqrt{x})$ and 5. This gives us $25 \sqrt{x}$. It's pretty straightforward – we're just multiplying the coefficient (5) with the term.
• Inner: Now, let's multiply the inner terms: 6 and $(5 \sqrt{x})$. This also results in $30 \sqrt{x}$. Again, we simply multiply the numbers together, keeping the square root term.
• Last: Finally, we multiply the last terms in each parenthesis: 6 and 5. This gives us 30. A simple multiplication, nothing tricky here!

So, after applying the FOIL method, we have: $25x + 25 \sqrt{x} + 30 \sqrt{x} + 30$. But we're not done yet! This is just the expanded form. We still need to simplify it to get our final answer. Think of it like baking a cake – we've mixed all the ingredients, but we need to bake it to get the final delicious product. In the next section, we'll focus on simplifying this expression by combining like terms. This is where we'll bring it all together and see the beauty of the simplified result. Keep going, you're doing great!

Simplifying the Expression

Now that we've expanded our expression using the FOIL method, we're sitting with $25x + 25 \sqrt{x} + 30 \sqrt{x} + 30$. The next step, and a crucial one, is to simplify this expression. Simplifying makes our answer cleaner, more concise, and easier to understand. It's like tidying up after a good cooking session – we want to leave a neat and organized space.

To simplify, we need to identify and combine like terms. Like terms are terms that have the same variable and exponent. In our expression, we have two terms with the square root of x: $25 \sqrt{x}$ and $30 \sqrt{x}$. These are like terms because they both contain $\sqrt{x}$. Think of $\sqrt{x}$ as a common unit – like saying we have 25 of these units and 30 of these units. It makes it easier to see how we can combine them.

So, let's combine those like terms. We add the coefficients (the numbers in front of the $\sqrt{x}$) together: 25 + 30. This gives us 55. Therefore, $25 \sqrt{x} + 30 \sqrt{x}$ simplifies to $55 \sqrt{x}$. We've effectively reduced two terms into one, making our expression simpler.

Now, let's rewrite our entire expression with this simplification: $25x + 55 \sqrt{x} + 30$. Notice that $25x$ and 30 don't have any like terms to combine with, so they stay as they are. This simplified expression is much cleaner and easier to work with. It's like decluttering your room – you've gotten rid of the unnecessary stuff and are left with the essentials. In this case, the essentials are the terms that cannot be further combined.

And that's it! We've successfully simplified our expression. We started with a somewhat complex expression and, through the application of the FOIL method and combining like terms, we've arrived at a much simpler form. In the next section, we'll recap our steps and provide the final answer, just to make sure everything's crystal clear. You've done an amazing job following along, and you're well on your way to mastering these types of problems!

Final Answer and Recap

Okay, guys, let's bring it all together and present our final answer. We started with the expression $(5 \sqrt{x}+6)(5 \sqrt{x}+5)$ and walked through the process of multiplying and simplifying it. We used the FOIL method to expand the expression and then combined like terms to get to our simplified form. It's been quite the journey, but we've arrived at our destination!

Our final simplified expression is: $25x + 55 \sqrt{x} + 30$. This is the result of multiplying $(5 \sqrt{x}+6)$ by $(5 \sqrt{x}+5)$ and simplifying the result. It's a neat and tidy expression that represents the product of the two original binomials.

Let's quickly recap the steps we took to get here. First, we understood the basics, including the distributive property (FOIL) and how square roots work. This foundational knowledge was crucial for tackling the problem effectively. Then, we applied the FOIL method, multiplying the First, Outer, Inner, and Last terms to expand the expression. This gave us a longer expression with several terms.

Next, we focused on simplifying the expression. We identified like terms, which were the terms containing $\sqrt{x}$, and combined them by adding their coefficients. This step is vital because it reduces the complexity of the expression and makes it easier to understand. Finally, we presented our simplified expression as the final answer.

Remember, practice makes perfect! The more you work through problems like this, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're part of the learning journey. Each time you work through a problem, you're reinforcing your understanding and building your skills. And who knows, maybe you'll even start to enjoy these types of problems! They're like little puzzles waiting to be solved.

So, there you have it! We've successfully multiplied and simplified an expression with square roots. Keep up the great work, and I'm sure you'll conquer any math challenge that comes your way. Until next time, happy calculating!