Simplifying Expressions Multiply And Simplify 4(√75 - √48)

Understanding the Problem

Alright guys, let's dive into this math problem where we need to multiply and then simplify the expression: 4(√75 - √48). At first glance, it might seem a bit intimidating with those square roots, but don't worry! We're going to break it down step by step, so it becomes super clear. Our main goal here is to make the expression as neat and simple as possible. This involves a couple of key techniques, focusing on simplifying those square roots before we even think about distributing the 4. Remember, the golden rule in math is often to simplify before you multiply – it makes life so much easier! We will use prime factorization to break down the numbers under the square roots, find pairs, and pull them out. Then, we'll combine like terms and finally multiply. Think of it as a mathematical puzzle where each step gets us closer to the final, elegant solution. So, let's roll up our sleeves and get started on this simplifying journey!

Breaking Down the Square Roots

So, the heart of simplifying this expression lies in tackling those square roots: √75 and √48. These numbers might seem scary, but we can simplify them using prime factorization. Prime factorization means breaking down a number into its prime factors – those prime numbers that multiply together to give you the original number. For √75, we can break down 75 into 3 × 25, and then further break down 25 into 5 × 5. So, 75 becomes 3 × 5 × 5. Now, remember that the square root of a number is a value that, when multiplied by itself, gives you the original number. So, √75 becomes √(3 × 5 × 5). We can pair up the two 5s, which gives us 5√3. This is because √(5 × 5) is simply 5, and the 3 stays under the square root since it doesn't have a pair. For √48, we can break it down into 2 × 24, then 24 into 2 × 12, 12 into 2 × 6, and finally 6 into 2 × 3. So, 48 becomes 2 × 2 × 2 × 2 × 3. Thus, √48 becomes √(2 × 2 × 2 × 2 × 3). We have two pairs of 2s here, so we can pull them out. This gives us 2 × 2 × √3, which simplifies to 4√3. See? It's like a treasure hunt for pairs! By breaking down the numbers into their prime factors, we can easily identify pairs and simplify the square roots. This is a crucial step in making the overall expression much more manageable.

Rewriting the Expression

Okay, so now that we've simplified the square roots, let's rewrite our original expression with the simplified forms. Remember, we started with 4(√75 - √48). We found that √75 simplifies to 5√3 and √48 simplifies to 4√3. So, we can substitute these values back into the expression. This gives us 4(5√3 - 4√3). See how much cleaner it already looks? We've gone from dealing with √75 and √48 to just dealing with √3. This is a massive step forward in simplifying the entire expression. The key here is substitution – replacing the original square roots with their simplified forms. This not only makes the expression easier to work with but also sets us up perfectly for the next step, which involves combining like terms and then finally distributing the 4. Think of it as swapping out clunky old parts for sleek new ones in a machine – the machine (our expression) will run much smoother now! This rewriting step is all about making the problem more approachable and setting the stage for the final simplification.

Combining Like Terms

Now, we're at the stage where we can combine like terms within the parentheses. Looking at our expression, 4(5√3 - 4√3), we see that we have two terms involving √3: 5√3 and -4√3. These are like terms because they both have the same square root part, which is √3. Combining like terms is just like combining apples and apples, or oranges and oranges. In this case, we're combining √3 terms. So, we simply subtract the coefficients (the numbers in front of the √3): 5 - 4. This gives us 1. Therefore, 5√3 - 4√3 simplifies to 1√3, which we usually just write as √3. Remember, when a coefficient is 1, we often don't explicitly write it, but it's still there. Our expression now looks even simpler: 4(√3). We've managed to condense the terms inside the parentheses into a single, clean term. This is a crucial step because it significantly reduces the complexity of the expression, making it much easier to handle. Combining like terms is a fundamental algebraic technique that helps us streamline expressions and get closer to the final simplified form. It's like tidying up a room – once everything is in its place, it's much easier to navigate!

Distributing the Multiplication

Alright, we're in the home stretch now! We've simplified the square roots, rewritten the expression, and combined like terms. Our expression currently looks like this: 4(√3). The final step is to distribute the multiplication. In this case, we simply need to multiply the 4 by the √3. Remember, multiplying a number by a square root is straightforward – you just write them next to each other. So, 4 * √3 becomes 4√3. And that's it! We've successfully multiplied and simplified the original expression. The final answer, 4√3, is in its simplest form. We can't simplify it any further because √3 is already in its simplest radical form, and there are no more like terms to combine. This final step of distributing the multiplication is often the quickest and most satisfying part of the process because it brings everything together. It's like putting the last piece of a puzzle in place – you can finally see the complete picture. So, to recap, we started with a seemingly complex expression, and through careful simplification and step-by-step manipulation, we arrived at a neat and tidy answer. Awesome job, guys!

Final Simplified Expression

So, after all the simplifying and multiplying, we've arrived at our final answer. Let's just highlight it one more time for clarity: The simplified form of the expression 4(√75 - √48) is 4√3. This is the most reduced and simplest form of the original expression. We've taken a somewhat complicated expression with square roots and transformed it into a much more manageable one. This process involved several key steps, including breaking down the square roots using prime factorization, rewriting the expression with the simplified square roots, combining like terms, and finally, distributing the multiplication. Each step played a crucial role in getting us to the final result. This journey from the initial complex expression to the final simplified form showcases the power of mathematical manipulation and simplification techniques. By applying these techniques, we can tackle even the most daunting-looking problems and make them much more approachable. Remember, simplifying expressions isn't just about getting the right answer; it's also about understanding the underlying mathematical principles and developing problem-solving skills. So, keep practicing, and you'll become a simplification pro in no time!