Rewrite \(\frac{10\sqrt[3]{z}}{2z^2}\) As \(k \cdot Z^n\)

Hey guys! Let's dive into rewriting the expression ${\frac{10\sqrt[3]{z}}{2z^2}}$ in the form ${k \cdot z^n}$, where we need to express the exponent as an integer, fraction, or exact decimal. No mixed numbers allowed! This might seem a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to follow. Think of it like turning a complicated math sentence into a simple, elegant equation. We're essentially translating from one math language to another, and once you get the hang of the rules, it's totally doable. Let's get started and make this expression shine!

Understanding the Basics

Before we jump into the problem, it's super important to have a solid grasp of a few key concepts. First, remember what exponents really mean. An exponent tells you how many times to multiply a number by itself. For example, ${z^2}$ means ${z \cdot z}$. But exponents can also be fractions, and that's where things get interesting. A fractional exponent represents a root. Specifically, ${z^{\frac{1}{n}}}$ is the same as the ${n}$-th root of ${z}$, written as ${\sqrt[n]{z}}$. So, ${z^{\frac{1}{3}}}$ is the cube root of ${z}$, or ${\sqrt[3]{z}}$. Understanding this connection between fractional exponents and roots is crucial for simplifying our expression. We also need to remember our exponent rules. One of the most important rules for this problem is how to handle division when you have the same base. When you divide exponents with the same base, you subtract the exponents. That is, ${\frac{z^m}{z^n} = z^{m-n}}$. This rule will be our best friend when we simplify the given expression. Finally, keep in mind that a constant multiplied by a variable raised to a power, like ${k \cdot z^n}$, is just a way of expressing how the variable changes. The constant ${k}$ is a scaling factor, and the exponent ${n}$ determines the rate of change. So, when we rewrite our expression in this form, we're essentially highlighting these two important aspects of the expression. This foundational knowledge will make the entire process much smoother and easier to understand.

Breaking Down the Expression

Okay, let's tackle the expression ${\frac{10\sqrt[3]{z}}{2z^2}}$ step by step. The first thing we can do is simplify the constants. We have ${\frac{10}{2}}$, which simplifies to 5. So, our expression now looks like ${\frac{5\sqrt[3]{z}}{z^2}}$. Next, we need to deal with that cube root. Remember that ${\sqrt[3]{z}}$ is the same as ${z^{\frac{1}{3}}}$. So, we can rewrite the expression as ${\frac{5z^{\frac{1}{3}}}{z^2}}$. Now we're getting somewhere! We have the same base, ${z}$, in both the numerator and the denominator. This is where our exponent rule for division comes into play. We know that ${\frac{z^m}{z^n} = z^{m-n}}$. In our case, ${m = \frac{1}{3}}$ and ${n = 2}$. So, we need to subtract 2 from ${\frac{1}{3}}$. Let's do that: ${\frac{1}{3} - 2 = \frac{1}{3} - \frac{6}{3} = -\frac{5}{3}}$. Therefore, ${\frac{z^{\frac{1}{3}}}{z^2} = z^{-\frac{5}{3}}}$. Putting it all together, our simplified expression is ${5z^{-\frac{5}{3}}}$. We've successfully rewritten the expression in the form ${k \cdot z^n}$, where ${k = 5}$ and ${n = -\frac{5}{3}}$. See? Not so scary when we break it down into manageable steps. We transformed a seemingly complex expression into a much simpler form by applying our knowledge of exponents and roots. This is a fantastic example of how understanding the fundamental rules of algebra can unlock the secrets of more complicated problems.

The Final Form

Alright, guys, we've done the heavy lifting and arrived at the simplified form of our expression: ${5z^{-\frac{5}{3}}}$. This is exactly in the ${k \cdot z^n}$ format that we were aiming for! Let's break down what this means. The coefficient, ${k}$, is 5. This tells us the scaling factor of the expression. The variable, ${z}$, is raised to the power of ${-\frac{5}{3}}$. This fractional exponent is super interesting because it combines a root and a negative exponent. Remember, a negative exponent means we're dealing with a reciprocal. So, ${z^{-\frac{5}{3}}}$ is the same as ${\frac{1}{z^{\frac{5}{3}}}}$. And what about that ${\frac{5}{3}}$? Well, we can think of it as ${z^{\frac{5}{3}} = (z^{\frac{1}{3}})^5 = (\sqrt[3]{z})^5}$. So, we're taking the cube root of ${z}$, raising it to the fifth power, and then taking the reciprocal of that result. This final form, ${5z^{-\frac{5}{3}}}$, is not just an answer; it's a powerful way to represent the relationship between the original expression and the variable ${z}$. It highlights the scaling factor and the combined effect of the root and the reciprocal. By rewriting the expression in this form, we've gained a much clearer understanding of its behavior. It's like having a secret decoder ring for math! We've taken a complex-looking expression and transformed it into something elegant and insightful. Awesome job, everyone!

Common Mistakes to Avoid

Now that we've successfully rewritten the expression, let's talk about some common pitfalls that people often encounter when tackling problems like this. One of the biggest mistakes is forgetting the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? You need to simplify exponents before you can multiply or divide. In our case, that means dealing with the cube root and the exponents before we simplify the fraction. Another common mistake is messing up the exponent rules, especially when dealing with fractional and negative exponents. For example, people sometimes forget that ${z^{-\frac{5}{3}}}$ is not the same as ${-z^{\frac{5}{3}}}$. The negative sign in the exponent means we're taking the reciprocal, not just making the whole term negative. Similarly, it's easy to mix up the rules for multiplying and dividing exponents. Remember, when you multiply terms with the same base, you add the exponents. When you divide, you subtract them. A classic mistake is to subtract the exponents when you should be adding them, or vice versa. Another pitfall is not simplifying the constants first. In our problem, we could simplify ${\frac{10}{2}}$ to 5 right away. Doing this early on makes the rest of the problem much cleaner and easier to manage. Finally, always double-check your work! It's easy to make a small arithmetic error, especially when dealing with fractions and negative signs. Take a moment to review each step and make sure everything adds up. By being aware of these common mistakes, you can avoid them and boost your confidence in tackling similar problems. Remember, practice makes perfect!

Practice Problems

To really nail this concept, let's try a few practice problems. Working through these will solidify your understanding and help you identify any areas where you might need a little extra review. Here are a few for you to try:

1. Rewrite ${\frac{15\sqrt{z}}{3z^3}}$ in the form ${k \cdot z^n}$.
2. Rewrite ${\frac{8z^4}{\sqrt[4]{z}}}$ in the form ${k \cdot z^n}$.
3. Rewrite ${\frac{4\sqrt[5]{z}}{2z^{\frac{2}{3}}}}$ in the form ${k \cdot z^n}$.

For each of these problems, remember to follow the same steps we used in the example. First, simplify the constants. Then, rewrite any roots as fractional exponents. Next, apply the exponent rules for division. Finally, express your answer in the form ${k \cdot z^n}$. Don't be afraid to take your time and work through each step carefully. The key is to understand the process, not just get the right answer. Once you've solved these problems, try making up your own! The more you practice, the more comfortable you'll become with these types of expressions. And remember, there's no substitute for practice when it comes to mastering math. So, grab a pencil and paper, and let's get to work! You've got this, I believe in you.

Conclusion

So, guys, we've successfully navigated the process of rewriting the expression ${\frac{10\sqrt[3]{z}}{2z^2}}$ in the form ${k \cdot z^n}$. We started by understanding the basic concepts of exponents and roots, then we broke down the expression step by step, and finally, we arrived at our answer: ${5z^{-\frac{5}{3}}}$. We also discussed common mistakes to avoid and tackled some practice problems to solidify our understanding. The key takeaway here is that complex expressions can be simplified by applying fundamental rules and breaking the problem down into manageable steps. Don't let the initial complexity intimidate you. With a solid understanding of the basics and a systematic approach, you can conquer even the most challenging problems. Remember, math is like a puzzle, and each step is like fitting a piece into place. The more pieces you fit, the clearer the picture becomes. And the feeling of solving a tough problem is incredibly rewarding. So, keep practicing, keep exploring, and keep pushing your boundaries. Math is not just about numbers and equations; it's about critical thinking, problem-solving, and the joy of discovery. Keep up the awesome work, and I'll see you in the next math adventure!