Simplify (3² × 5²) ⋅ 25 ⋅ 49 / (35² ⋅ 90²)

Introduction

Hey guys! Today, we're going to dive into simplifying a complex mathematical expression. Expressions like these might seem daunting at first, but with a few tricks and a good understanding of mathematical principles, we can break them down into something much more manageable. Our mission is to simplify the expression M = (3² × 5²) ⋅ 25 ⋅ 49 / (35² ⋅ 90²). This involves understanding the order of operations, prime factorization, and how to cancel out common factors. So, let's put on our math hats and get started! We'll take it step by step, ensuring that each part is crystal clear. By the end of this, you’ll not only be able to simplify this expression but also tackle similar problems with confidence. Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them logically. So, grab a pen and paper, and let's embark on this mathematical adventure together. This initial breakdown is crucial because it sets the stage for the simplification process. We're essentially laying the groundwork by identifying the key components and the operations involved.

Breaking Down the Expression

To effectively simplify the expression M, let's first break it down into its constituent parts. The expression is M = (3² × 5²) ⋅ 25 ⋅ 49 / (35² ⋅ 90²). We have several components here: squares of numbers, multiplication, and division. Our initial step involves understanding each part individually before we combine them. Let's start by looking at the squares and how they affect the numbers. For instance, 3² means 3 multiplied by itself, which equals 9. Similarly, 5² is 5 times 5, giving us 25. These initial calculations help us reduce the complexity of the expression. Next, we have 25 and 49, which are also perfect squares (5² and 7², respectively). Recognizing these perfect squares is crucial for simplification because it allows us to express them in their prime factors easily. This is where understanding prime factorization comes into play. Prime factorization is the process of breaking down a number into its prime number components. For example, 25 can be expressed as 5 × 5, and 49 can be expressed as 7 × 7. This breakdown is vital because it helps us identify common factors in the numerator and the denominator, which we can later cancel out. By breaking down each component individually, we make the overall expression less intimidating and more approachable. Now, let's move on to the denominator, where we have 35² and 90². These terms also need to be broken down into their prime factors. This step-by-step approach ensures that we don’t miss any critical details and sets us up for efficient simplification. Remember, the key is to take a complex problem and make it simpler by addressing each part methodically. This initial breakdown is the foundation upon which we will build our simplification strategy.

Prime Factorization

The heart of simplifying this expression lies in prime factorization. Understanding this concept is crucial, guys. Prime factorization is the process of breaking down a number into its prime number components. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). Breaking down the numbers into their prime factors allows us to see the common elements in both the numerator and the denominator, which we can then cancel out. Let’s start with the numerator. We have (3² × 5²) ⋅ 25 ⋅ 49. We already know 3² is 3 × 3 and 5² is 5 × 5. The number 25 can be expressed as 5 × 5 (or 5²), and 49 can be expressed as 7 × 7 (or 7²). So, the numerator becomes (3 × 3 × 5 × 5) ⋅ (5 × 5) ⋅ (7 × 7). This detailed breakdown helps us visualize all the prime factors involved. Now, let's tackle the denominator, which is 35² ⋅ 90². First, we need to express 35 and 90 in their prime factors before squaring them. The number 35 can be factored into 5 × 7, so 35² is (5 × 7)². This means 35² is (5 × 7) × (5 × 7), which simplifies to 5 × 5 × 7 × 7. Next, let’s factor 90. The number 90 can be divided into 9 × 10. The number 9 can be further factored into 3 × 3, and 10 can be factored into 2 × 5. Therefore, 90 = 2 × 3 × 3 × 5. Squaring 90 means (2 × 3 × 3 × 5)², which is (2 × 3 × 3 × 5) × (2 × 3 × 3 × 5). This expands to 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5. Writing both the numerator and the denominator in their prime factor forms is a critical step. It transforms the expression into a form where we can easily identify and cancel out common factors. This process makes the simplification much more straightforward and less prone to errors. By understanding prime factorization, we’re setting ourselves up for success in simplifying the expression. It’s like having a secret code that unlocks the solution. So, make sure you’re comfortable with this concept before moving on. This technique is super helpful not just for this problem but for many other mathematical challenges as well.

Rewriting the Expression with Prime Factors

Now that we've broken down each component into its prime factors, let’s rewrite the entire expression. This step is essential for visualizing how the numbers can be simplified. Our original expression was M = (3² × 5²) ⋅ 25 ⋅ 49 / (35² ⋅ 90²). After performing prime factorization, we can rewrite this expression with all the numbers expressed in their prime forms. The numerator (3² × 5²) ⋅ 25 ⋅ 49 becomes (3 × 3 × 5 × 5) ⋅ (5 × 5) ⋅ (7 × 7). This is simply replacing each number with its prime factors. Similarly, the denominator 35² ⋅ 90² becomes (5 × 5 × 7 × 7) ⋅ (2 × 2 × 3 × 3 × 3 × 3 × 5 × 5). Again, we've substituted each number with its corresponding prime factors. So, our expression M now looks like this:

M = (3 × 3 × 5 × 5 × 5 × 5 × 7 × 7) / (5 × 5 × 7 × 7 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5)

This might look a bit intimidating at first glance, but don't worry! The beauty of this form is that it allows us to see exactly which factors are common in both the numerator and the denominator. These common factors are the key to simplifying the expression. By rewriting the expression in this way, we’ve essentially laid out all the pieces of the puzzle. The next step is to identify and cancel out these common factors. Think of it like decluttering – we’re removing the unnecessary parts to reveal the simpler form underneath. This step is not just about rewriting; it's about transforming the expression into a format that’s easier to work with. It’s a crucial bridge between the complex original expression and its simplified form. Remember, the more organized we are in our approach, the smoother the simplification process will be. So, take your time, double-check your prime factorizations, and make sure everything is correctly transcribed. This thoroughness will pay off in the next step when we start canceling out the factors. This transformation is a game-changer in simplifying complex expressions.

Cancelling Out Common Factors

Alright, guys, this is where the magic happens! Now that we have the expression rewritten with all the prime factors, we can start canceling out the common factors from both the numerator and the denominator. This step is all about identifying and eliminating matching pairs of prime numbers. Looking at our rewritten expression:

M = (3 × 3 × 5 × 5 × 5 × 5 × 7 × 7) / (5 × 5 × 7 × 7 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5)

We can see several factors that appear in both the numerator and the denominator. Let’s start with the smaller numbers. We have 3 × 3 in the numerator and 3 × 3 × 3 × 3 in the denominator. We can cancel out two pairs of 3s from both, leaving us with two 3s in the denominator. Next, we have 5 × 5 × 5 × 5 in the numerator and 5 × 5 × 5 × 5 in the denominator. All these 5s can be completely canceled out. This is a significant simplification, as it removes a substantial part of the expression. Moving on, we have 7 × 7 in the numerator and 7 × 7 in the denominator. These also cancel each other out completely. After canceling out the common factors, our expression looks much simpler. The numerator has been reduced to just 1 (since everything has been canceled out), and the denominator now has 2 × 2 × 3 × 3 remaining. The canceled factors essentially divide out to 1, so we're left with the remaining factors. This process of cancellation is the heart of simplification. It's like pruning a tree – we’re removing the excess to reveal the essential structure. Each cancellation makes the expression more manageable and brings us closer to the final answer. This step requires careful attention to detail. It’s crucial to ensure that you’re only canceling out factors that appear in both the numerator and the denominator. A mistake here can throw off the entire simplification process. So, take your time, double-check your cancellations, and make sure you’re being precise. This methodical approach is what transforms a complex-looking expression into a simple, elegant result. By the end of this step, you’ll feel a real sense of accomplishment as you see the expression shrinking down to its core components. This cancellation process is a powerful tool in mathematics, and mastering it will help you tackle a wide range of problems with confidence.

Final Simplification

Okay, we're in the home stretch now! After canceling out all the common factors, we’re left with a much simpler expression. Let’s see what we have. Our expression after cancellation looks like this:

M = 1 / (2 × 2 × 3 × 3)

Now, all that’s left is to multiply the remaining numbers in the denominator. We have 2 × 2, which equals 4, and 3 × 3, which equals 9. So, the denominator becomes 4 × 9. Multiplying 4 and 9 gives us 36. Therefore, the simplified expression is:

M = 1 / 36

And there we have it! We’ve successfully simplified the original complex expression into a fraction. This final step is about putting everything together and arriving at the answer. It's the culmination of all our hard work – breaking down the expression, performing prime factorization, rewriting with prime factors, and canceling out common elements. The result, 1/36, is the simplest form of the original expression. This final simplification not only gives us the answer but also highlights the power of methodical problem-solving. By following a clear, step-by-step approach, we transformed a seemingly daunting problem into a straightforward calculation. This skill is invaluable, not just in mathematics but in many areas of life. It’s about taking complexity and turning it into simplicity. Remember, the journey to the solution is just as important as the solution itself. Each step we took – from breaking down the expression to canceling out factors – taught us something valuable. And now, seeing the final simplified form, we can appreciate the elegance and efficiency of mathematical principles. So, congratulations! You’ve successfully simplified a complex expression, and you’ve gained a deeper understanding of the process. This newfound confidence will serve you well in future mathematical challenges. Keep practicing, keep exploring, and remember that every complex problem can be simplified with the right approach.

Conclusion

So, there you have it, guys! We've successfully simplified the expression M = (3² × 5²) ⋅ 25 ⋅ 49 / (35² ⋅ 90²) down to 1/36. We started with a complex-looking expression and, by breaking it down step by step, we made it manageable and understandable. The key techniques we used were breaking down the expression into smaller parts, performing prime factorization, rewriting the expression with prime factors, canceling out common factors, and then doing the final simplification. Each step was crucial in getting us to the final answer. This exercise demonstrates the power of a methodical approach to problem-solving. By taking a complex problem and breaking it down into smaller, more manageable steps, we can tackle almost anything. Remember, math isn’t just about getting the right answer; it’s about understanding the process and the underlying principles. The skills you've learned here – prime factorization, simplification, and methodical problem-solving – are transferable to many other areas of mathematics and beyond. Keep practicing these techniques, and you’ll find that simplifying complex expressions becomes second nature. Math can be challenging, but it’s also incredibly rewarding. The feeling of solving a complex problem and arriving at the correct answer is a great confidence booster. So, keep exploring, keep learning, and most importantly, keep having fun with math! Whether you're tackling algebra, calculus, or any other mathematical challenge, the principles we’ve discussed here will serve you well. Simplifying expressions is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts and applications. So, take pride in your accomplishment, and keep pushing your mathematical boundaries. This journey of simplifying the expression is a testament to the fact that with patience, persistence, and the right tools, any mathematical challenge can be overcome. And who knows, maybe you'll even start to enjoy the process of simplification! This is just one step in your mathematical journey, and there are many more exciting challenges and discoveries ahead.