Simplifying Algebraic Expressions What Is The Product Of 5k/6 * 3/2k^3

Let's dive into understanding the product of the expression $\frac{5k}{6} \cdot \frac{3}{2k^3}$. This problem falls squarely into the realm of mathematics, specifically dealing with algebraic expressions and simplification. Guys, don't worry if it looks a bit intimidating at first! We're going to break it down step by step, making it super easy to grasp. The core concept here is multiplying fractions that involve variables, and then simplifying the resulting expression. This involves handling both the numerical coefficients and the variable terms with their exponents. It’s like combining ingredients in a recipe – we'll multiply the numerators (the top parts of the fractions) together and the denominators (the bottom parts) together. Then, we'll simplify by canceling out common factors, much like tidying up after cooking! This kind of problem is super common in algebra, and mastering it opens the door to more complex equations and concepts. So, let’s get started and see how we can simplify this expression to its simplest form. We’ll take it one step at a time, making sure each part makes perfect sense. Think of it as a puzzle – each step is a piece, and when we put them all together, we get the solution! Remember, mathematics isn't about memorizing formulas, it's about understanding the process and the 'why' behind each step. So let's unlock this puzzle together!

Breaking Down the Expression

Alright, let's get into the nitty-gritty and really dissect this expression: $\frac{5k}{6} \cdot \frac{3}{2k^3}$. The first thing we want to do is think about what each part represents. We have two fractions here, and they're being multiplied. Remember, when we multiply fractions, we multiply the numerators together to get the new numerator, and we multiply the denominators together to get the new denominator. So, in this case, we'll be multiplying $5k$ by $3$ and $6$ by $2k^3$. It’s kind of like saying we have two separate "things" and we want to combine them into one. Think of the $k$ as a placeholder, a variable that represents some unknown value. It's super important to remember that it follows the same rules of mathematics as regular numbers. The exponents, like the $3$ in $k^3$, tell us how many times we're multiplying $k$ by itself (so $k^3$ is $k \cdot k \cdot k$). We'll need to keep this in mind when we simplify later. This step is all about setting the stage. We're identifying the components and understanding the operation we need to perform. It's like reading the instructions before assembling a piece of furniture – we need to know what we're working with before we start putting things together! The better we understand the individual parts, the smoother the simplification process will be. So, we've got our two fractions, we know we're multiplying them, and we're ready to move on to the next step: actually performing the multiplication.

Multiplying the Fractions

Okay, now for the fun part – actually multiplying these fractions! As we discussed, we're going to multiply the numerators together and the denominators together. So, we have $(5k) \cdot (3)$ in the numerator and $(6) \cdot (2k^3)$ in the denominator. Let's break this down further. $(5k) \cdot (3)$ is the same as $5 \cdot k \cdot 3$. We can rearrange this because multiplication is commutative (meaning the order doesn't matter) to get $5 \cdot 3 \cdot k$, which simplifies to $15k$. So, the new numerator is $15k$. Now, let's tackle the denominator: $(6) \cdot (2k^3)$. This is the same as $6 \cdot 2 \cdot k^3$, which simplifies to $12k^3$. So, our new denominator is $12k^3$. This means our expression now looks like $\frac{15k}{12k^3}$. We've successfully multiplied the fractions together! This is a big step, guys. Think of it like building the main structure of a house – we've got the walls up! But we're not done yet. Now we need to simplify, which is like adding the finishing touches to make everything look polished and perfect. Multiplying fractions is a fundamental skill in mathematics, and you've just nailed it. This step highlights the importance of understanding the rules of multiplication and how they apply to variables and coefficients. Now we're ready to move on to the final, and often most satisfying, step: simplification!

Simplifying the Result

Alright, let's simplify! We've got $\frac{15k}{12k^3}$, and our goal is to make this look as clean and simple as possible. This is where we look for common factors in the numerator and the denominator that we can cancel out. First, let's focus on the numbers: 15 and 12. What's the greatest common factor (GCF) of 15 and 12? If you're thinking 3, you're spot on! We can divide both 15 and 12 by 3. So, 15 divided by 3 is 5, and 12 divided by 3 is 4. This means we can rewrite our fraction as $\frac{5k}{4k^3}$. Now, let's tackle the variable part: we have $k$ in the numerator and $k^3$ in the denominator. Remember that $k^3$ means $k \cdot k \cdot k$. So, we have one $k$ in the numerator and three $k$s in the denominator. We can cancel out one $k$ from both the top and the bottom. This is like subtracting exponents: $k^1$ (which is just $k$) divided by $k^3$ is $k^{1-3} = k^{-2}$, which is the same as $\frac{1}{k^2}$. So, after canceling out the $k$, our fraction becomes $\frac{5}{4k^2}$. And there you have it! We've simplified the expression to its simplest form. This final step is super satisfying because it's where we see all our hard work pay off. We've taken a seemingly complex expression and boiled it down to something much more manageable. Simplifying is a crucial skill in mathematics, as it allows us to work with expressions more easily and to see the underlying relationships more clearly. We've gone from multiplying fractions to identifying common factors and canceling them out, showcasing the interconnectedness of different mathematical concepts. So, congratulations, guys, you've successfully simplified the expression!

Final Answer

So, after all that awesome work, we've arrived at our final answer. The simplified form of the expression $\frac{5k}{6} \cdot \frac{3}{2k^3}$ is $\frac{5}{4k^2}$. This means that if you were to plug in any value for $k$ (except 0, because we can't divide by zero!), the original expression and this simplified expression would give you the same result. Pretty cool, huh? This whole process has been a journey through the world of algebraic fractions, and we've covered some really important concepts along the way. We've talked about multiplying fractions, identifying common factors, and simplifying expressions with variables and exponents. These are skills that will serve you well in all sorts of mathematical adventures to come. Remember, mathematics is like a language – the more you practice, the more fluent you become. And simplifying expressions is like speaking that language clearly and concisely. It's about taking something complex and making it understandable. So, keep practicing, keep exploring, and keep having fun with mathematics! You've got this, guys! And now you can confidently say that you know how to find the product and simplify expressions like this one. High five!