Solving (6√(5v) + 4√(7v))(-3√(5v) + 5√(7)) Simply

Hey guys! Ever stumbled upon a math problem that looks like it's speaking a different language? Well, today, we're going to break down one of those problems and make it super easy to understand. We're talking about expressions with square roots, specifically this one: $(6 \sqrt{5 v}+4 \sqrt{7 v})(-3 \sqrt{5 v}+5 \sqrt{7})$. Sounds intimidating, right? Don't worry, we'll tackle it together, step by step, and you'll be a pro in no time!

Diving into the Distributive Property

So, how do we even begin to solve something like this? The key here is the distributive property, which might sound fancy, but it's really just a method of multiplying each term inside the first set of parentheses by each term in the second set. Think of it like making sure everyone at a party gets a piece of the cake – you've got to distribute it evenly! In this case, we have two terms in the first parenthesis $(6 \sqrt{5 v}$ and $4 \sqrt{7 v})$ and two terms in the second parenthesis $(-3 \sqrt{5 v}$ and $5 \sqrt{7})$. We will multiply each term in the first parenthesis by each term in the second parenthesis, leading to four multiplication operations in total. This is often remembered using the acronym FOIL, which stands for First, Outer, Inner, Last. It's a handy way to remember the order in which to multiply the terms. However, the distributive property is the underlying principle, so understanding that is more important than just memorizing FOIL. Let's break it down:

1. First: Multiply the first terms in each parenthesis: $(6 \sqrt{5v}) * (-3 \sqrt{5v})$
2. Outer: Multiply the outer terms: $(6 \sqrt{5v}) * (5 \sqrt{7})$
3. Inner: Multiply the inner terms: $(4 \sqrt{7v}) * (-3 \sqrt{5v})$
4. Last: Multiply the last terms in each parenthesis: $(4 \sqrt{7v}) * (5 \sqrt{7})$

By following this methodical approach, we ensure that we account for every possible pairing of terms, setting the stage for simplification and ultimately solving the expression. Each of these multiplications will produce a term, and by combining these terms appropriately, we can arrive at the simplified form of the original expression. So, let's put on our mathematical thinking caps and dive into each of these multiplications to see what they reveal!

Multiplying the Terms: A Step-by-Step Guide

Okay, let's get our hands dirty and actually do the multiplication. This is where things start to get interesting! Remember, we're going to take each pair of terms we identified using the distributive property and multiply them together. The trick here is to remember the rules for multiplying square roots: you multiply the numbers outside the square roots together, and you multiply the numbers inside the square roots together. Let's take it one step at a time:

1. First Terms: $(6 \sqrt{5v}) * (-3 \sqrt{5v})$

• Multiply the coefficients (the numbers outside the square root): 6 * -3 = -18
• Multiply the terms inside the square root: $\sqrt{5v} * \sqrt{5v} = \sqrt{(5v)(5v)} = \sqrt{25v^2}$
• Simplify the square root: $\sqrt{25v^2} = 5v$ (assuming v is non-negative)
• Combine the results: -18 * 5v = -90v

2. Outer Terms: $(6 \sqrt{5v}) * (5 \sqrt{7})$

• Multiply the coefficients: 6 * 5 = 30
• Multiply the terms inside the square root: $\sqrt{5v} * \sqrt{7} = \sqrt{(5v)(7)} = \sqrt{35v}$
• Combine the results: $30\sqrt{35v}$

3. Inner Terms: $(4 \sqrt{7v}) * (-3 \sqrt{5v})$

• Multiply the coefficients: 4 * -3 = -12
• Multiply the terms inside the square root: $\sqrt{7v} * \sqrt{5v} = \sqrt{(7v)(5v)} = \sqrt{35v^2}$
• Simplify the square root: $\sqrt{35v^2} = v\sqrt{35}$ (assuming v is non-negative)
• Combine the results: $-12v\sqrt{35}$

4. Last Terms: $(4 \sqrt{7v}) * (5 \sqrt{7})$

• Multiply the coefficients: 4 * 5 = 20
• Multiply the terms inside the square root: $\sqrt{7v} * \sqrt{7} = \sqrt{(7v)(7)} = \sqrt{49v}$
• Simplify the square root: $\sqrt{49v} = 7\sqrt{v}$
• Combine the results: $20 * 7\sqrt{v} = 140\sqrt{v}$

See? It's like a puzzle where each piece fits together perfectly. We've now successfully multiplied all the terms, and we're one giant leap closer to the final answer. The next step involves gathering all these individual results and seeing if we can simplify them further. This is where we'll look for like terms and combine them, making our expression as neat and tidy as possible. So, hang tight, we're on the home stretch!

Combining Like Terms and Simplifying

Alright, we've done the hard work of multiplying everything out. Now comes the satisfying part: simplifying! We've got four terms now: -90v, $30\sqrt{35v}$, $-12v\sqrt{35}$, and $140\sqrt{v}$. The goal here is to identify any terms that are similar and can be combined. Remember, like terms have the same variable parts – in this case, the same square root part and the same variable part.

Looking at our terms, we notice something interesting: $30\sqrt{35v}$ and $-12v\sqrt{35}$ both have $\sqrt{35}$ and 'v' in them, but one has 'v' inside the square root and the other has 'v' outside. Because of this, they are not like terms and cannot be combined directly. However, let's rewrite $-12v\sqrt{35}$ as $-12\sqrt{35v^2}$ to highlight the difference in the placement of 'v'.

So, it seems there are no like terms to combine in this particular expression. This means our expression is already in its simplest form after the multiplication and individual simplifications of square roots. Sometimes, that's just how it goes! Not every problem has a dramatic simplification at the end, and that's perfectly okay. It's important to recognize when you've reached the simplest form, even if it's not as concise as you might have hoped.

This highlights a crucial aspect of problem-solving in mathematics: knowing when you've reached the end of the road. It's tempting to keep manipulating an expression, hoping for further simplification, but sometimes the most important step is recognizing that you've done all you can. In our case, we've multiplied, simplified individual terms, and checked for like terms. Since there are none, we can confidently say we've reached the simplified form of the expression.

The Final Result: Unveiling the Solution

So, after all that multiplying, simplifying, and checking, what's our final answer? Well, it turns out the simplified form of the expression $(6 \sqrt{5 v}+4 \sqrt{7 v})(-3 \sqrt{5 v}+5 \sqrt{7})$ is:

-90v + 30√(35v) - 12v√(35) + 140√v

Yep, it's a bit of a mouthful, but that's perfectly alright! Sometimes math problems don't have super neat and tidy answers, and that's part of what makes them interesting. What's important is that we followed the correct steps, applied the distributive property properly, and simplified each term as much as possible. We've arrived at the correct solution, even if it looks a little complex.

This result showcases the importance of careful and methodical work in mathematics. Each step, from applying the distributive property to simplifying square roots, played a crucial role in arriving at the final answer. Skipping steps or making careless errors along the way could easily have led to an incorrect result. By taking our time and paying attention to detail, we were able to navigate the complexities of the expression and confidently reach the solution.

Why This Matters: Real-World Applications

Now, you might be thinking, "Okay, that's cool, but when am I ever going to use this in real life?" That's a fair question! While you might not be multiplying expressions with square roots every day, the underlying principles we used to solve this problem are incredibly valuable in many different fields. The distributive property, simplifying expressions, and breaking down complex problems into smaller, manageable steps are all skills that can be applied in various situations.

Think about engineering, for example. Engineers often deal with complex equations and formulas when designing structures, machines, or electrical circuits. They need to be able to manipulate these equations, simplify them, and solve for unknown variables. The same skills we used to tackle our square root expression are essential for engineers to do their jobs effectively. Similarly, in computer science, simplifying complex algorithms and optimizing code involves breaking down problems into smaller parts and applying logical rules, just like we did in our math problem.

Even in fields like finance and economics, the ability to manipulate and simplify complex expressions is crucial. Financial analysts use mathematical models to predict market trends, assess risk, and make investment decisions. Economists use similar models to analyze economic data, forecast economic growth, and develop policy recommendations. The underlying mathematical principles are the same, even if the context is different.

So, while you might not see square roots popping up in your daily life, the problem-solving skills you develop by tackling these types of math problems are incredibly valuable and transferable to a wide range of fields. Learning to think critically, break down complex problems, and apply logical rules are skills that will serve you well no matter what path you choose in life. Keep practicing, keep challenging yourself, and you'll be amazed at what you can achieve!

Conclusion: You've Conquered the Square Roots!

Awesome! You've made it to the end, and you've successfully tackled a challenging math problem involving square roots. Give yourself a pat on the back – you've earned it! We started with a seemingly complex expression, $(6 \sqrt{5 v}+4 \sqrt{7 v})(-3 \sqrt{5 v}+5 \sqrt{7})$, and broke it down step by step. We used the distributive property, carefully multiplied each term, simplified the square roots, and looked for like terms to combine. And even though our final answer wasn't the simplest-looking thing in the world, we arrived at the correct solution by following the rules and being meticulous in our work.

Remember, the key to success in math (and in life!) is to break down big problems into smaller, more manageable steps. Don't be afraid to tackle something that looks intimidating at first. With a little bit of patience, a clear understanding of the rules, and a willingness to work through the details, you can conquer anything. Keep practicing, keep learning, and keep pushing yourself to new challenges. You've got this!