Simplifying 18√2250 A Step-by-Step Solution

Hey guys! Today, we're diving deep into a fascinating mathematical problem: simplifying the expression 18√2250. This might seem daunting at first, but trust me, with a step-by-step approach, we can conquer this together. So, grab your thinking caps, and let's embark on this mathematical journey!

Deconstructing the Square Root: A Primer

Before we tackle the main problem, let's brush up on the fundamentals of square roots. 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 * 3 = 9. The symbol '√' denotes the square root. Understanding this concept is crucial for simplifying expressions involving square roots.

The key to simplifying square roots lies in identifying perfect square factors within the number under the radical (the √ symbol). A perfect square is a number that can be obtained by squaring an integer (a whole number). Examples of perfect squares include 4 (2 * 2), 9 (3 * 3), 16 (4 * 4), and so on. By extracting these perfect square factors, we can simplify the square root expression. This is because the square root of a perfect square is simply the integer that was squared to obtain it. For example, √16 = 4, because 16 is the perfect square of 4. This principle allows us to rewrite the original square root in a simpler, more manageable form, making calculations easier and the overall expression more understandable. The process of identifying and extracting perfect square factors is a cornerstone of simplifying square root expressions and is essential for solving more complex mathematical problems involving radicals.

Simplifying √2250: The Heart of the Matter

Now, let's focus on the core of our problem: simplifying √2250. Our goal is to find the largest perfect square that divides evenly into 2250. This might seem challenging, but there's a systematic way to approach it. We can start by breaking down 2250 into its prime factors. Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (examples: 2, 3, 5, 7, 11, etc.). By finding the prime factors, we can easily identify any perfect square factors.

So, let's find the prime factorization of 2250. We can start by dividing 2250 by the smallest prime number, 2. We find that 2250 = 2 * 1125. Now, we need to factor 1125. It's not divisible by 2, so we move on to the next prime number, 3. We find that 1125 = 3 * 375. Continuing this process, we find that 375 = 3 * 125, and 125 = 5 * 25, and finally, 25 = 5 * 5. Putting it all together, the prime factorization of 2250 is 2 * 3 * 3 * 5 * 5 * 5. This representation is incredibly useful because it allows us to clearly see the components of 2250 and identify any repeated factors that could form perfect squares.

Now, let's rewrite the prime factorization, grouping the factors into pairs: 2250 = 2 * (3 * 3) * (5 * 5) * 5. Notice that we have a pair of 3s (3 * 3) and a pair of 5s (5 * 5). These pairs represent perfect squares: 3 * 3 = 9 (which is 3 squared) and 5 * 5 = 25 (which is 5 squared). This is a crucial observation because it allows us to simplify the square root. We can rewrite √2250 as √(2 * 3² * 5² * 5). Remember, the square root of a product is the product of the square roots, so we can further break this down as √2 * √3² * √5² * √5. The square root of 3 squared (√3²) is simply 3, and the square root of 5 squared (√5²) is simply 5. Therefore, we have √2250 = 3 * 5 * √(2 * 5) = 15√10. This simplification is a significant step forward, as we've reduced the complex √2250 into a much more manageable form, 15√10. This process highlights the power of prime factorization in simplifying square roots and making mathematical expressions easier to work with.

Multiplying by 18: The Final Step

We've successfully simplified √2250 to 15√10. Now, we need to multiply this result by 18, as the original expression was 18√2250. This step is relatively straightforward and involves multiplying the whole number parts together. So, we have 18 * (15√10). Remember, we can only multiply the numbers outside the square root symbol. This gives us (18 * 15)√10.

Let's calculate 18 * 15. We can break this down if it helps: 18 * 10 = 180, and 18 * 5 = 90. Adding these together, we get 180 + 90 = 270. So, 18 * 15 = 270. Therefore, 18 * (15√10) = 270√10. This is our final simplified answer. We've taken the original expression, 18√2250, and through careful simplification of the square root and multiplication, we've arrived at 270√10. This process demonstrates how breaking down complex problems into smaller, more manageable steps can lead to a clear and concise solution. The ability to simplify such expressions is a valuable skill in mathematics, allowing us to work with numbers more efficiently and accurately.

Therefore, the answer to 18√2250 is 270√10.

I hope this step-by-step explanation has helped you understand how to simplify expressions like 18√2250. Remember, the key is to break down the problem into smaller, more manageable steps. Keep practicing, and you'll become a pro at simplifying square roots in no time!