Simplifying The Expression: A Step-by-Step Guide

Hey guys! Today, we're going to dive into simplifying the algebraic expression $\frac{3 y^3}{7 y^4}+\frac{15 y}{14 y^3}$. It might look a bit intimidating at first, but don't worry! We'll break it down step by step, making it super easy to understand. So, let's get started!

Understanding the Basics

Before we jump into the simplification, let's quickly review some fundamental concepts. We need to be comfortable with fractions, exponents, and how to combine like terms. Think of this as our toolkit for tackling this problem. Remember, mathematics is like building with LEGOs – each piece (or concept) fits together to create something awesome. Our goal here is to make the complex look simple, and that starts with mastering the fundamentals.

Fractions and their Simplification

Fractions, at their core, represent a part of a whole. They consist of two main components: the numerator (the top number) and the denominator (the bottom number). Simplifying fractions means reducing them to their simplest form. We achieve this by finding the greatest common factor (GCF) of both the numerator and the denominator and then dividing both by it. This process doesn't change the value of the fraction; it just expresses it in a more concise manner. Imagine you have a pizza cut into 8 slices, and you eat 4 slices. That’s 4/8 of the pizza, but you could also say you ate 1/2 of the pizza. Both fractions represent the same amount, just in different forms. This is the magic of simplifying fractions, and it’s crucial for what we’re about to do.

The Laws of Exponents

Exponents are a shorthand way of representing repeated multiplication. For instance, $y^3$ means $y * y * y$. Understanding the laws of exponents is crucial for simplifying expressions like ours. One of the most relevant rules for this problem is the quotient rule, which states that when dividing like bases, you subtract the exponents. Mathematically, this is expressed as $\frac{a^m}{a^n} = a^{m-n}$. This rule is going to be our best friend when we deal with the variables in our expression. Another key concept is that any variable raised to the power of 0 is 1 (except for 0 itself). These rules might seem abstract now, but they become incredibly useful when you start applying them. Think of exponents as the gears in a machine; they help us efficiently manipulate and simplify expressions.

Combining Like Terms

Combining like terms is like sorting your laundry – you group similar items together. In algebraic expressions, like terms are those that have the same variable raised to the same power. For example, $3y^2$ and $5y^2$ are like terms because they both have $y^2$, but $3y^2$ and $5y^3$ are not like terms because the exponents are different. To combine like terms, you simply add or subtract their coefficients (the numbers in front of the variables). This process helps to tidy up expressions and make them easier to work with. It’s like organizing your toolbox; keeping similar tools together makes it easier to find what you need and simplifies your work.

Step-by-Step Simplification

Okay, with our toolkit ready, let's tackle the expression $\frac{3 y^3}{7 y^4}+\frac{15 y}{14 y^3}$ step-by-step. We'll break it down into manageable chunks to make sure we don't miss anything.

Step 1: Simplify Each Fraction Separately

The first thing we're going to do is simplify each fraction individually. This means we'll look at $\frac{3 y^3}{7 y^4}$ and $\frac{15 y}{14 y^3}$ separately before we try to add them together. This approach helps to minimize confusion and allows us to focus on one part of the problem at a time. It's like clearing your desk before starting a new task; it reduces distractions and makes the process smoother.

Simplifying $\frac{3 y^3}{7 y^4}$

Let's start with the first fraction, $\frac{3 y^3}{7 y^4}$. To simplify this, we need to look at both the numerical coefficients and the variables. The numerical part is $\frac{3}{7}$, which is already in its simplest form since 3 and 7 have no common factors other than 1. Now, let’s focus on the variables. We have $\frac{y^3}{y^4}$. Remember the quotient rule of exponents? It says that $\frac{a^m}{a^n} = a^{m-n}$. Applying this rule, we get $\frac{y^3}{y^4} = y^{3-4} = y^{-1}$. A negative exponent means we can rewrite the expression as a fraction: $y^{-1} = \frac{1}{y}$. So, putting it all together, $\frac{3 y^3}{7 y^4}$ simplifies to $\frac{3}{7y}$. See how we broke it down? Simple, right?

Simplifying $\frac{15 y}{14 y^3}$

Now, let's move on to the second fraction, $\frac{15 y}{14 y^3}$. Again, we'll tackle the numerical coefficients and the variables separately. The numerical part is $\frac{15}{14}$, which also doesn’t simplify further since 15 and 14 have no common factors other than 1. For the variables, we have $\frac{y}{y^3}$. Using the quotient rule again, we get $\frac{y}{y^3} = y^{1-3} = y^{-2}$. This means we can rewrite it as $\frac{1}{y^2}$. Combining these results, $\frac{15 y}{14 y^3}$ simplifies to $\frac{15}{14y^2}$. We’re halfway there! Each step is like solving a mini-puzzle, and when we put them all together, the big picture becomes clear.

Step 2: Find a Common Denominator

Now that we've simplified each fraction, we have $\frac{3}{7y} + \frac{15}{14y^2}$. To add these fractions, we need a common denominator. Think of it like needing the same unit to measure things; you can’t directly add inches and centimeters without converting them first. Finding a common denominator allows us to combine the numerators in a meaningful way. The least common denominator (LCD) is the smallest multiple that both denominators share. In this case, our denominators are $7y$ and $14y^2$.

Determining the Least Common Denominator (LCD)

To find the LCD, we need to consider both the numerical coefficients and the variable parts of the denominators. For the numerical coefficients, we have 7 and 14. The least common multiple (LCM) of 7 and 14 is 14 because 14 is a multiple of 7. For the variables, we have $y$ and $y^2$. The least common multiple of $y$ and $y^2$ is $y^2$ because $y^2$ is divisible by $y$. Therefore, the least common denominator (LCD) for our fractions is $14y^2$. This is the denominator we’ll use to combine our fractions. Finding the LCD is like finding the right tool for the job; it sets the stage for smooth sailing ahead.

Converting the Fractions to the Common Denominator

Now that we have our common denominator, $14y^2$, we need to convert each fraction to have this denominator. For the first fraction, $\frac{3}{7y}$, we need to multiply both the numerator and the denominator by the same factor that will turn $7y$ into $14y^2$. That factor is $2y$ because $7y * 2y = 14y^2$. So, we multiply the fraction like this: $\frac{3}{7y} * \frac{2y}{2y} = \frac{6y}{14y^2}$. For the second fraction, $\frac{15}{14y^2}$, we already have the correct denominator, so we don’t need to change it. It stays as $\frac{15}{14y^2}$. Now, our expression looks like this: $\frac{6y}{14y^2} + \frac{15}{14y^2}$. We’re one step closer to the finish line!

Step 3: Add the Fractions

With the fractions now having a common denominator, we can finally add them together! Adding fractions with a common denominator is like adding slices of the same pie; you just add the numerators while keeping the denominator the same. In our case, we have $\frac{6y}{14y^2} + \frac{15}{14y^2}$. To add these, we simply add the numerators: $6y + 15$. The denominator stays the same, so we have $\frac{6y + 15}{14y^2}$. This is the sum of our fractions, but we're not quite done yet. We need to check if we can simplify the result further.

Step 4: Simplify the Resulting Fraction

Our expression is currently $\frac{6y + 15}{14y^2}$. To simplify this, we need to look for common factors in the numerator and the denominator. First, let’s examine the numerator, $6y + 15$. We can factor out a common factor of 3 from both terms: $3(2y + 5)$. Now, our fraction looks like this: $\frac{3(2y + 5)}{14y^2}$. Next, we look at the denominator, $14y^2$. The prime factors of 14 are 2 and 7. The numerator has a factor of 3, and the term $(2y + 5)$ does not share any common factors with 14 or $y^2$. Therefore, there are no further common factors we can cancel out. So, our final simplified expression is $\frac{3(2y + 5)}{14y^2}$. We did it! We took a complex-looking expression and simplified it to its core.

Final Answer

So, after all those steps, the simplified form of the expression $\frac{3 y^3}{7 y^4}+\frac{15 y}{14 y^3}$ is $\frac{3(2y + 5)}{14y^2}$.

Conclusion

There you have it, guys! Simplifying algebraic expressions might seem daunting at first, but by breaking it down into smaller, manageable steps, it becomes much easier. Remember, the key is to understand the basic concepts like fractions, exponents, and combining like terms. With practice, you’ll become a pro at simplifying even the most complex expressions. Keep practicing, and you’ll be amazed at what you can achieve. Happy simplifying!