Multiplying & Simplifying (z+8)/8 * (z+6)/z: A Step-by-Step Guide

by Andrew McMorgan 66 views

Hey guys! Ever get tangled up in those tricky rational expressions? Today, we're diving deep into how to multiply and simplify one specific example: (z+8)/8 * (z+6)/z. We'll break it down step-by-step, so you can confidently tackle similar problems. So, grab your pencils, and let's get started!

Understanding Rational Expressions

Before we jump into the problem, let's quickly recap what rational expressions are. Simply put, a rational expression is a fraction where the numerator and the denominator are polynomials. Polynomials, in turn, are expressions containing variables raised to non-negative integer powers, combined with constants and arithmetic operations (addition, subtraction, multiplication). Think of expressions like z+8, 8, z+6, and z – these are all polynomials. Putting two polynomials in a fraction form creates our rational expression, like the ones we're dealing with today. Working with these expressions often involves simplifying them, which means reducing them to their simplest form without changing their value. This can involve factoring, canceling common factors, and, as we'll see today, multiplying them together. When multiplying rational expressions, the key is to multiply the numerators together and multiply the denominators together. Then, we simplify the resulting fraction, which might involve some algebraic manipulation. This process is similar to how you multiply regular fractions, but with the added complexity of dealing with variables and polynomials. So, keep your algebra skills sharp, and let's get ready to simplify!

Step-by-Step Breakdown: Multiplying the Numerators

Okay, let's dive into the first part of our problem: multiplying the numerators. Our expression is (z+8)/8 * (z+6)/z, so the numerators we're dealing with are (z+8) and (z+6). To multiply these two binomials, we'll use the distributive property, which is often remembered by the acronym FOIL: First, Outer, Inner, Last. This method ensures we multiply each term in the first binomial by each term in the second binomial. First, we multiply the first terms of each binomial: z * z = z². Next, we multiply the outer terms: z * 6 = 6z. Then, we multiply the inner terms: 8 * z = 8z. Finally, we multiply the last terms: 8 * 6 = 48. Now, we add all these products together: z² + 6z + 8z + 48. The final step is to combine any like terms. In this case, we have 6z and 8z, which add up to 14z. So, our final product for the numerators is z² + 14z + 48. Remember this result, as it's a crucial part of our final simplified expression. Multiplying binomials like this is a fundamental skill in algebra, so mastering it will make simplifying rational expressions much easier. Next, we'll tackle the denominators!

Multiplying the Denominators: A Simpler Task

Now that we've conquered the numerators, let's move on to the denominators of our expression (z+8)/8 * (z+6)/z. This part is actually a bit more straightforward. We need to multiply the denominators, which are 8 and z. This is a simple multiplication: 8 * z = 8z. That's it! No need for FOIL or any fancy techniques here. We've got our product of the denominators: 8z. This result will form the denominator of our combined fraction. It's worth noting that sometimes, the denominators might be more complex expressions, requiring the distributive property or other algebraic techniques. However, in this case, we're fortunate to have a simple multiplication. This highlights the importance of recognizing different types of expressions and knowing which techniques to apply. With our numerators and denominators now multiplied, we're ready to combine them into a single fraction and move on to the simplification process. So, let's put these pieces together!

Combining the Results: Forming the New Fraction

Alright, we've done the hard work of multiplying the numerators and the denominators separately. Now it's time to combine our results and form the new fraction. Remember, we found that the product of the numerators, (z+8) * (z+6), is z² + 14z + 48. And the product of the denominators, 8 * z, is 8z. To form our new fraction, we simply place the product of the numerators over the product of the denominators. This gives us the fraction: (z² + 14z + 48) / 8z. This fraction represents the result of multiplying the original rational expressions. However, we're not quite done yet! The final step is to simplify this fraction as much as possible. This usually involves looking for common factors in the numerator and the denominator that we can cancel out. In the next section, we'll focus on the crucial process of simplifying this fraction. So, we've come a long way, but the journey to complete simplification continues! Stay tuned!

Simplifying the Resulting Fraction: Factoring and Cancelling

Now comes the exciting part: simplifying the fraction we obtained, which is (z² + 14z + 48) / 8z. To simplify, we need to look for common factors in the numerator and the denominator that we can cancel out. The first step in this process is usually to factor the numerator. Factoring means rewriting the quadratic expression z² + 14z + 48 as a product of two binomials. We need to find two numbers that add up to 14 (the coefficient of the z term) and multiply to 48 (the constant term). After some thought, we can see that the numbers 6 and 8 fit the bill, since 6 + 8 = 14 and 6 * 8 = 48. Therefore, we can factor the numerator as (z + 6)(z + 8). Now our fraction looks like this: ((z + 6)(z + 8)) / 8z. Next, we look for common factors between the numerator and the denominator. In this case, there are no common factors that we can directly cancel out. The denominator is 8z, which has factors of 8 and z. The numerator has factors of (z + 6) and (z + 8), none of which match the factors in the denominator. This means that the fraction is already in its simplest form. So, the simplified form of the expression is (z² + 14z + 48) / 8z or ((z + 6)(z + 8)) / 8z. Understanding how to factor quadratic expressions and identify common factors is crucial for simplifying rational expressions. It's a skill that will serve you well in many areas of algebra and beyond.

Final Answer and Key Takeaways

So, we've reached the end of our journey! We started with the expression (z+8)/8 * (z+6)/z, and after multiplying the numerators and denominators, and then simplifying, we arrived at the final answer: (z² + 14z + 48) / 8z or ((z + 6)(z + 8)) / 8z. Since there are no common factors between the numerator and the denominator, this is the simplest form of the expression. Let's recap the key steps we took:

  1. Multiply the numerators: We used the FOIL method to multiply (z+8) and (z+6), resulting in z² + 14z + 48.
  2. Multiply the denominators: We multiplied 8 and z, resulting in 8z.
  3. Combine the results: We formed the fraction (z² + 14z + 48) / 8z.
  4. Simplify the fraction: We attempted to factor the numerator, but found no common factors to cancel with the denominator. The fraction was already in its simplest form.

This problem highlights some important concepts in working with rational expressions. Firstly, multiplying rational expressions involves multiplying the numerators and the denominators separately. Secondly, simplifying rational expressions often requires factoring and canceling common factors. And lastly, it's important to recognize when an expression is already in its simplest form. Hopefully, this step-by-step guide has helped you understand how to multiply and simplify rational expressions. Keep practicing, and you'll become a pro in no time! Remember, algebra can be fun, guys! Keep exploring and tackling those problems!