Multiplying Rational Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of rational expressions and figuring out how to multiply them. It's like a fun puzzle, and I'm here to break it down step-by-step. We'll start with the problem: What is the product of the rational expressions?

$\left(\frac{5 y}{9}\right)\left(\frac{x+3}{4 x}\right)$

And we'll go through the process to nail the correct answer from the multiple choices. Let's get started!

Understanding Rational Expressions: The Basics

Before we get our hands dirty with the multiplication, let's make sure we're all on the same page about what rational expressions are. Think of them as fractions, but instead of just numbers, we have algebraic expressions in the numerator and the denominator. These expressions can involve variables, constants, and all sorts of mathematical operations. The key is that we're dealing with fractions where the parts are algebraic. This is important to understand because many students struggle when they have to do things with variables. It's like they think it's a completely different subject, but it's not!

To make this super clear, a rational expression is essentially a fraction where the numerator and/or denominator is/are polynomials. Polynomials are expressions with variables and constants combined using addition, subtraction, and multiplication. Now, remember the most important rule: you can't divide by zero! So, when we're working with rational expressions, we always have to keep in mind the values that would make the denominator equal to zero. These values are excluded from the domain of the expression because they are undefined. It's like a secret rule that keeps everything in check. Keep this in mind because we'll need to remember it later.

So, why do we even care about rational expressions? Well, they pop up all over the place! We use them to model real-world situations, solve equations, and simplify complex problems. For example, they're super helpful in physics, engineering, and economics. Once you get the hang of it, you'll see they are very useful. I know it sounds boring, but trust me. So, understanding how to work with these expressions is a fundamental skill in algebra and beyond. It's like having a key that unlocks a bunch of other mathematical doors. You'll be using this in all sorts of different scenarios.

The Multiplication Process: How to Multiply Rational Expressions

Alright, let's get down to the nitty-gritty and see how to multiply rational expressions. The good news is that the process is fairly straightforward. It's basically the same as multiplying regular fractions, but with algebraic expressions. The general idea is:

1. Multiply the numerators: Take the expressions in the numerators of each fraction and multiply them together. If you're using this with variables, this is an important step.
2. Multiply the denominators: Do the same thing for the denominators – multiply those expressions together.
3. Simplify (if possible): This is where things get interesting. After multiplying, you'll likely have a new rational expression. Check to see if you can simplify it by canceling out any common factors in the numerator and the denominator. This often involves factoring the expressions. Don't worry, we'll go through all of this in detail.

Let's apply this to our problem. We are going to multiply $\left(\frac{5 y}{9}\right)\left(\frac{x+3}{4 x}\right)$.

First, multiply the numerators. In our case, we have $5y$ and $(x + 3)$. So, the numerator of our resulting fraction will be $5y * (x + 3)$.

Second, multiply the denominators. We have $9$ and $4x$, so the denominator of our resulting fraction will be $9 * 4x$.

So far, we have $\frac{5y(x+3)}{9(4x)}$. Now we must simplify and solve.

Step-by-Step Solution: Multiplying Our Rational Expressions

Let's put the process into action with our given problem. We need to find the product of: $\left(\frac{5 y}{9}\right)\left(\frac{x+3}{4 x}\right)$.

Step 1: Multiply the Numerators.

Multiply the numerators together: $5y * (x + 3)$. This gives us $5y * x + 5y * 3$. So, the result is $5xy + 15y$.

Step 2: Multiply the Denominators.

Multiply the denominators together: $9 * 4x$. This gives us $36x$.

Step 3: Write the Resulting Rational Expression.

Now we put the new numerator and denominator together to form the new rational expression: $\frac{5xy+15y}{36x}$.

Step 4: Check for Simplification.

Is there anything we can simplify in the expression $\frac{5xy+15y}{36x}$? Well, we need to check if we can factor anything out. In the numerator, we can factor out a $5y$, but that doesn't help us simplify anything with the denominator because there are no common factors. So, the expression is already in its simplest form.

Now, let's go back and see our multiple-choice options:

A. $\frac{5 x y+3}{36 x}$ B. $\frac{5 x y+15 y}{36 x}$ C. $\frac{5 y+x+3}{4 x+9}$ D. $\frac{15 x y}{36 x}$

From our step-by-step process, we know the correct answer must be $\frac{5xy+15y}{36x}$.

Analyzing the Answer Choices: Finding the Right Match

Now, let's take a closer look at the multiple-choice options to see which one matches our solution. We already know the correct answer. But, let's pretend we don't, so that we can demonstrate how to find the answer. Remember our calculated answer: $\frac{5xy+15y}{36x}$.

• Option A: $\frac{5 x y+3}{36 x}$ – This option is incorrect because the numerator doesn't match. It has a $+3$, but the actual numerator should have a $+15y$.
• Option B: $\frac{5 x y+15 y}{36 x}$ – Bingo! This is the correct answer. The numerator is $5xy + 15y$, and the denominator is $36x$, which matches our calculated result.
• Option C: $\frac{5 y+x+3}{4 x+9}$ – This option is way off. The numerator and denominator are completely different from what we calculated.
• Option D: $\frac{15 x y}{36 x}$ – This option is incorrect because the numerator is missing the $15y$. Also, we can further simplify it and it still would not be the correct option.

So, by carefully multiplying the rational expressions and comparing our result with the options, we can confidently pick the right answer, which is option B. It’s super important to go through each step carefully.

Conclusion: Mastering Rational Expression Multiplication

Alright, folks, we've successfully navigated the multiplication of rational expressions! We started with a problem, broke it down into simple steps, and ended up with the correct answer. You can see how easy it is! Remember the key takeaways:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the resulting expression if possible.

With practice, you'll become a pro at this. Keep practicing, and don't be afraid to ask for help if you get stuck. You've got this! Now go forth and conquer those rational expressions. Until next time, keep the math vibes strong! I hope this helps you out, guys!