Simplifying The Expression: 7/(6u^3) + 3/(4u^2) - A Step-by-Step Guide
Hey guys! Today, we're diving into simplifying a fractional expression that might look a little intimidating at first, but trust me, it's totally manageable. We're going to tackle the expression 7/(6u^3) + 3/(4u^2). Think of it like combining two different recipes – we need to find a common ground before we can mix them. So, let's break it down step by step and make it super easy to understand.
Understanding the Basics of Fraction Simplification
Before we jump into the specific problem, let's quickly recap the core idea behind simplifying fractions. When you're adding or subtracting fractions, the golden rule is that you need a common denominator. Imagine trying to add apples and oranges – you can't directly do it until you have a common unit, like "fruit." The same applies to fractions. The denominator is the bottom part of the fraction (like the '6u^3' and '4u^2' in our case), and we need to make them the same before we can combine the numerators (the top parts, '7' and '3').
Finding the common denominator involves a bit of number-crunching and variable-wrangling, but it's a skill that will come in handy time and time again in algebra and beyond. We're essentially looking for the smallest expression that both denominators can divide into evenly. This is called the least common multiple (LCM) of the denominators. Once we have that, we can adjust the fractions so they both have the same denominator, and then we can finally add them together. It's like finding the perfect recipe conversion so you can combine all your ingredients seamlessly. So, let's get started on the nitty-gritty of our specific expression!
Step 1: Finding the Least Common Denominator (LCD)
Okay, so the first thing we need to do when simplifying 7/(6u^3) + 3/(4u^2) is to find the least common denominator (LCD). This is like finding the smallest shared ingredient in our recipe analogy. To do this, we need to look at the denominators, which are 6u^3 and 4u^2. We'll break this down into two parts: the numerical coefficients (6 and 4) and the variable terms (u^3 and u^2).
Numerical Coefficients
Let's start with the numbers. We need to find the least common multiple (LCM) of 6 and 4. Think of the multiples of each number: Multiples of 6 are 6, 12, 18, 24, and so on. Multiples of 4 are 4, 8, 12, 16, and so on. The smallest number that appears in both lists is 12. So, the LCM of 6 and 4 is 12. This means our common denominator will have a '12' in it. We're one step closer to cracking this!
Variable Terms
Now, let's tackle the variable terms: u^3 and u^2. When finding the LCM of variables with exponents, we take the highest power of the variable that appears in either term. In this case, we have u^3 and u^2. The highest power is u^3, so that's what we'll use in our LCD. It's like making sure you have enough of the main ingredient to cover all parts of the dish. So, putting it all together, the least common denominator (LCD) for 6u^3 and 4u^2 is 12u^3. This is the magic number we need to transform our fractions and get them ready for addition. On to the next step!
Step 2: Adjusting the Fractions
Now that we've found the least common denominator (LCD) for 7/(6u^3) + 3/(4u^2), which is 12u^3, the next step is to adjust each fraction so that they both have this denominator. Think of it like scaling up a recipe – we need to make sure we adjust all the ingredients proportionally to keep the flavor the same. To do this, we'll multiply both the numerator and the denominator of each fraction by whatever it takes to get the denominator to be 12u^3.
Adjusting the First Fraction
Let's start with the first fraction, 7/(6u^3). We already have 6u^3 in the denominator, and we want to get to 12u^3. To do this, we need to multiply the denominator by 2 (since 6 * 2 = 12). Remember, whatever we do to the denominator, we also have to do to the numerator to keep the fraction equivalent. So, we'll multiply both the numerator and the denominator by 2: (7 * 2) / (6u^3 * 2) = 14 / (12u^3). Now our first fraction is ready to go!
Adjusting the Second Fraction
Next up is the second fraction, 3/(4u^2). Here, we have 4u^2 in the denominator, and we need to get to 12u^3. To get the numerical coefficient right, we need to multiply 4 by 3 (since 4 * 3 = 12). For the variable part, we have u^2, and we need u^3, so we need to multiply by u (since u^2 * u = u^3). So, we'll multiply both the numerator and the denominator by 3u: (3 * 3u) / (4u^2 * 3u) = 9u / (12u^3). Great! Both our fractions now have the same denominator, which means we're ready for the fun part – adding them together. Let's move on to the next step!
Step 3: Adding the Fractions
Alright, we've reached the point where we can finally add our fractions together! We've adjusted 7/(6u^3) + 3/(4u^2) so that they both have the same denominator: 12u^3. Our expression now looks like this: 14/(12u^3) + 9u/(12u^3). This is like having all our ingredients prepped and ready to mix in the bowl. Adding fractions with a common denominator is super straightforward.
When the denominators are the same, we simply add the numerators and keep the denominator the same. It’s like saying, “If I have 14 slices of pizza out of 12u^3 total slices, and then I add 9u slices, how many slices do I have in total?” So, we add the numerators: 14 + 9u. The denominator stays the same: 12u^3. This gives us the combined fraction (14 + 9u) / (12u^3). We're almost there!
Now, before we declare victory, we need to check if we can simplify the fraction any further. This means looking for any common factors between the numerator (14 + 9u) and the denominator (12u^3). In this case, 14 and 9 don't share any common factors, and the numerator doesn't have a 'u' term that we can cancel out with the denominator. So, it looks like we've reached the end of the road for simplification. This is our final answer!
Final Answer
So, after all the steps, the simplified form of the expression 7/(6u^3) + 3/(4u^2) is (14 + 9u) / (12u^3). You did it! We took a potentially tricky problem and broke it down into manageable steps. Remember, simplifying fractions is all about finding that common denominator and then combining the numerators. With a bit of practice, you'll be simplifying expressions like a pro. Keep up the great work, guys!
