Simplifying Expressions With Exponents A Step By Step Guide

Hey guys! Let's dive into simplifying this algebraic expression: $x^0 y z^{-3}$. I know it might look a bit intimidating at first, but trust me, we'll break it down step by step, making it super easy to understand. We'll go through the fundamental rules of exponents and apply them to each part of the expression. By the end of this article, you'll not only know the correct answer but also grasp the underlying principles. So, grab your thinking caps, and let’s get started!

Understanding the Basics of Exponents

Before we jump into the problem, let's quickly recap the basics of exponents. Exponents, also known as powers, indicate how many times a number (the base) is multiplied by itself. For example, in the expression $a^n$, 'a' is the base, and 'n' is the exponent. This means we multiply 'a' by itself 'n' times. Now, there are a few key rules we need to remember. First, any number raised to the power of 0 is equal to 1. That is, $a^0 = 1$, provided that 'a' is not zero. This might seem a bit strange, but it's a fundamental rule that simplifies many algebraic expressions. Second, a negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, a^{-n} = rac{1}{a^n}. This rule is crucial for dealing with expressions like $z^{-3}$ in our problem. These rules form the backbone of exponent manipulation, and mastering them will make simplifying expressions like a breeze. We'll see how these rules come into play as we tackle the given expression. Understanding these foundational concepts ensures we can confidently approach more complex problems later on. So, let's keep these rules in mind and move on to simplifying our expression!

Breaking Down the Expression $x^0 y z^{-3}$

Okay, let's get our hands dirty with the expression $x^0 y z^{-3}$. To simplify this, we'll apply the exponent rules we just discussed, one term at a time. First up, we have $x^0$. Remember the rule that anything to the power of 0 is 1? So, $x^0$ simply becomes 1. Easy peasy! Next, we have the 'y' term. This one is straightforward since it's already in its simplest form with an exponent of 1 (which we usually don't write explicitly). So, 'y' remains as 'y'. Now, let's tackle the last term, $z^{-3}$. This is where the negative exponent rule comes into play. Recall that $a^{-n} = \frac{1}{a^n}$. Applying this rule, $z^{-3}$ becomes $\frac{1}{z^3}$. See? We're making progress! Now we've simplified each part of the expression. $x^0$ is 1, 'y' remains 'y', and $z^{-3}$ transforms into $\frac{1}{z^3}$. The next step is to put these simplified pieces back together. This is where we'll combine our results to get the final simplified expression. Stay with me, guys; we're almost there!

Combining the Simplified Terms

Alright, we've broken down the expression $x^0 y z^{-3}$ into its individual components and simplified each one. Now it's time to bring them all back together. We found that $x^0 = 1$, 'y' remains 'y', and $z^{-3} = \frac{1}{z^3}$. So, let's substitute these back into the original expression: $1 * y * \frac{1}{z^3}$. Multiplying these terms together is pretty straightforward. 1 multiplied by anything remains the same, so we have $y * \frac{1}{z^3}$. Now, we can rewrite this as a single fraction. Multiplying 'y' by $\frac{1}{z^3}$ gives us $\frac{y}{z^3}$. And that's it! We've successfully simplified the expression. This final form is much cleaner and easier to understand than the original. We've taken an expression with a zero exponent and a negative exponent and transformed it into a simple fraction. It's all about applying those exponent rules systematically. Now, let's take a look at our answer choices and see which one matches our simplified expression. This step is crucial to ensure we've arrived at the correct solution and haven't made any sneaky mistakes along the way. So, keep your eyes peeled as we compare our result with the given options!

Identifying the Correct Answer

Okay, we've simplified $x^0 y z^{-3}$ to $\frac{y}{z^3}$. Now, let's match this with the given answer choices. We have:

A. $\frac{x y}{z^3}$ B. $\frac{x}{y z^3}$ C. $\frac{y}{z^3}$ D. $\frac{1}{y z^3}$

Looking at these options, we can clearly see that option C, $\frac{y}{z^3}$, matches our simplified expression. Options A, B, and D have extra terms or are in a different form, so they're not the correct answer. Option A includes an 'x' term in the numerator, which we know isn't correct because $x^0$ simplifies to 1. Option B has 'x' in the numerator and 'y' in the denominator, neither of which is present in our simplified form. Option D has 'y' in the denominator, which is also incorrect. Therefore, by carefully comparing our result with the options, we can confidently say that option C is the correct answer. It's always a good practice to double-check your work and ensure that your simplified expression aligns perfectly with the answer choice you select. So, give yourselves a pat on the back, guys! We've successfully simplified the expression and identified the correct answer. But let's not stop here; let's recap what we've learned to solidify our understanding.

Recap and Key Takeaways

Awesome job, guys! We've successfully simplified the expression $x^0 y z^{-3}$ and found the correct answer. Let's take a moment to recap the key steps and takeaways from this problem. First, we started by understanding the fundamental rules of exponents. We refreshed our knowledge that any number raised to the power of 0 equals 1, and we reviewed the rule for negative exponents, which tells us that $a^{-n} = \frac{1}{a^n}$. These rules are the bread and butter of simplifying exponential expressions. Next, we broke down the given expression into its individual components: $x^0$, 'y', and $z^{-3}$. We simplified each term separately, applying the exponent rules where necessary. This step-by-step approach made the problem much more manageable. Then, we combined the simplified terms. We replaced $x^0$ with 1 and $z^{-3}$ with $\frac{1}{z^3}$, and then we multiplied the terms together to get our simplified expression, $\frac{y}{z^3}$. Finally, we compared our simplified expression with the given answer choices and confidently identified option C as the correct answer. The key takeaway here is that simplifying expressions with exponents involves applying the rules systematically and breaking the problem down into smaller, more manageable parts. Remember, practice makes perfect! The more you work with exponents, the more comfortable you'll become with these rules. So, keep practicing, and you'll be simplifying expressions like a pro in no time! And there you have it, guys! We've not only solved the problem but also reinforced our understanding of exponent rules. Keep up the great work!

Final Answer: The final answer is $\boxed{C}$