Solving (1/4 X^(2/7)) * (5/14 X^(-2/3)) * (3/7 X^(9/2)) A Step By Step Guide

Hey guys! Today, we're going to break down how to solve the expression (1/4 * x^(2/7)) * (5/14 * x^(-2/3)) * (3/7 * x^(9/2)). This looks intimidating at first glance, but don’t worry! We'll tackle it step by step to make it super clear. Our main goal is to simplify this expression, and we'll do that by using the rules of exponents and basic arithmetic. So, let's dive right in and get this sorted!

Understanding the Basics

Before we jump into the solution, let's quickly recap some fundamental concepts that we'll be using. This will make the entire process much smoother. First off, remember the rules of exponents. When you multiply terms with the same base, you add the exponents. For example, x^a * x^b = x^(a+b). This is crucial for simplifying our expression. Also, don't forget how to multiply fractions. You simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For instance, (a/b) * (c/d) = (ac) / (bd). Lastly, it's super important to handle negative exponents correctly. A term with a negative exponent can be rewritten as its reciprocal with a positive exponent. That is, x^(-a) = 1 / x^a. Keeping these basics in mind will help us navigate through the problem with ease. Alright, let’s keep these tools handy as we start dissecting the expression!

Step 1: Grouping Like Terms

The first thing we’re going to do is group the like terms together. This makes the expression look less cluttered and easier to handle. We have three main components in our expression: the numerical fractions and the terms with x raised to various powers. So, let's rewrite the expression by grouping the fractions together and the x terms together. This will look something like this: (1/4 * 5/14 * 3/7) * (x^(2/7) * x^(-2/3) * x^(9/2)). By doing this, we've separated the constants from the variables, which sets us up perfectly for the next steps. It’s all about making the problem more manageable, right? Grouping like terms is a simple yet powerful technique that helps us keep things organized and prevents us from making silly mistakes. Now, we’re ready to tackle each group separately. Let’s start with the numerical fractions!

Step 2: Multiplying the Numerical Fractions

Now that we’ve grouped our like terms, let’s focus on simplifying the numerical fractions. We have (1/4) * (5/14) * (3/7). To multiply fractions, you simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. So, we have (1 * 5 * 3) / (4 * 14 * 7). Multiplying the numerators gives us 15, and multiplying the denominators gives us 392. Thus, our fraction becomes 15/392. But wait, we’re not done yet! It’s always a good idea to see if we can simplify the fraction further. In this case, 15 and 392 don’t share any common factors other than 1, so the fraction 15/392 is in its simplest form. Yay! We’ve successfully multiplied and simplified the numerical part of our expression. Next up, we’ll tackle the x terms. Let's keep the momentum going!

Step 3: Simplifying the Exponents of x

Alright, let’s move on to the fun part – simplifying the exponents of x. We have x^(2/7) * x^(-2/3) * x^(9/2). Remember the rule we talked about earlier? When multiplying terms with the same base, you add the exponents. So, we need to add the exponents 2/7, -2/3, and 9/2. This means we have x^((2/7) + (-2/3) + (9/2)). Now, to add these fractions, we need to find a common denominator. The least common multiple (LCM) of 7, 3, and 2 is 42. So, we'll convert each fraction to have a denominator of 42. Let's do it: (2/7) = (2 * 6) / (7 * 6) = 12/42, (-2/3) = (-2 * 14) / (3 * 14) = -28/42, and (9/2) = (9 * 21) / (2 * 21) = 189/42. Now we can add them: 12/42 - 28/42 + 189/42 = (12 - 28 + 189) / 42 = 173/42. So, the exponent of x simplifies to 173/42. That wasn’t so bad, right? We’ve just handled the exponents like pros! Now, let’s bring it all together.

Step 4: Combining the Simplified Terms

Okay, we're in the home stretch now! We've simplified both the numerical fractions and the x terms separately. We found that the numerical part simplifies to 15/392, and the exponent of x simplifies to 173/42. Now, we just need to combine these two parts to get our final simplified expression. Remember, we had (15/392) * (x^(173/42)). So, putting it all together, our final simplified expression is (15/392)x^(173/42). And there you have it! We’ve successfully simplified the original expression. Give yourself a pat on the back! It might have looked daunting at first, but breaking it down step by step made it totally manageable. Let's recap the whole process to make sure we’ve got it down pat.

Step 5: Final Answer

So, after all that work, our final answer is (15/392)x^(173/42). We took a complicated-looking expression and simplified it using basic rules of exponents and fraction multiplication. Remember, the key is to break down the problem into smaller, manageable steps. First, we grouped the like terms. Then, we multiplied the numerical fractions and simplified them. Next, we tackled the exponents by finding a common denominator and adding them up. Finally, we combined our simplified terms to arrive at the final answer. This step-by-step approach can be used for many different types of math problems. Don’t be intimidated by complex expressions; just take it one step at a time, and you’ll get there! Now, let’s do a quick recap of the main steps we followed.

Conclusion: Key Takeaways

Alright, guys! Let’s wrap things up with some key takeaways from this exercise. We started with a seemingly complex expression and simplified it by following a structured approach. The most important thing to remember is to break down the problem. Don’t try to do everything at once. Instead, group like terms, simplify each group separately, and then combine the results. We used the rules of exponents extensively, so make sure you’re comfortable with those. Adding exponents when multiplying terms with the same base, dealing with negative exponents, and finding common denominators are crucial skills. Also, remember the basic arithmetic of multiplying fractions: multiply the numerators and the denominators. Finally, always simplify your answer as much as possible. Look for common factors to reduce fractions and ensure your exponents are in their simplest form. By following these steps, you’ll be well-equipped to tackle similar problems. Keep practicing, and you’ll become a pro at simplifying expressions in no time!

I hope this step-by-step guide has made solving this type of expression much clearer for you guys. Remember, math isn’t about memorizing formulas; it’s about understanding the process. Keep practicing, and you’ll master it in no time! If you have any questions or want to tackle more problems together, just let me know. Keep up the great work!