Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem involving algebraic expressions. We're going to break down how to simplify a given expression into a form with positive exponents. This is super important in algebra, so pay attention! We'll start with the basics and go through each step clearly. Ready to get started?

Understanding the Problem: What We're Dealing With

Okay, so the problem gives us an expression, let's call it A. This expression includes variables like x, y, and z, each with different exponents, and some constants. The initial form of A has some negative exponents. Our main goal is to rewrite A so that all the exponents are positive. Why do we want to do this? Well, it's often considered the simplest form of the expression, and it's easier to work with in further calculations. Plus, it looks neater! The key here is to remember the rules of exponents. Things like how to handle negative exponents, how to combine terms with the same base, and how to simplify fractions with exponents. Let's break it down step by step, making sure it all clicks. Before we begin, it's a good idea to quickly recap some of the basic exponent rules. The rules are the foundation we build upon. When we have a term with a negative exponent, like x⁻², we can rewrite it as 1/x². When multiplying terms with the same base, we add the exponents (e.g., x² * x*³ = x⁵). And when dividing terms with the same base, we subtract the exponents (e.g., x⁵ / x² = x³). These rules are going to be our best friends throughout this process. We will also look at how to simplify fractions. Simplifying fractions is the same as dealing with the exponents. We want to make it as simple as possible, reducing the number of terms and making sure the exponents are positive.

The Original Expression

The expression A is given as:

A = (2 * x⁻² * y⁶ * z¹) / (100 * x⁻⁸ * y⁻¹ * z⁻³)

Our mission? To transform this into an equivalent form where all exponents are positive. It might look a little intimidating at first, but trust me, we can do this together. We'll take it slow and make sure every move makes sense. The initial expression has a lot going on. There's a fraction, there are variables with negative exponents, and there are some constants. Breaking it into smaller parts will help us to not get overwhelmed. We will start with the constants, then look at the x terms, then the y terms, and finally, the z terms. Keeping each part separate makes it easy to track and avoid mistakes. The more practice you get at these types of problems, the better you'll be. So, don't feel discouraged if it seems tough at first. We're building a skill, and like any skill, it improves with practice. We will start by separating the constant terms from the variables. Then, we will focus on simplifying the variables, using the exponent rules. In the end, all the negative signs will be gone, and all the exponents will be positive.

Step-by-Step Simplification: Turning Negatives into Positives

Alright, let's get down to business! We're going to go through this expression step-by-step. We'll start with the constant terms and then move on to the variables. Remember, the goal is to end up with positive exponents only. So, let's do this!

Simplifying the Constants

First, let's deal with the constants: 2 and 100. We can simplify the fraction 2/100, which becomes 1/50. So, we now have:

A = (1/50) * (x⁻² * y⁶ * z¹) / (x⁻⁸ * y⁻¹ * z⁻³)

This is already looking a little cleaner, right? Simplifying the fraction 2/100 gives us 1/50. Keep this in mind because it changes the whole expression and makes things easier. Remember, simplifying constants is often the first step in these kinds of problems. By simplifying the constants first, you are setting the stage to handle the rest of the variables more easily. Don't skip this step! It will make your life much simpler down the road. Now, the expression looks more manageable. We reduced the numbers and can now focus on the more interesting part: variables.

Dealing with the x Variables

Now, let's focus on the x variables. We have x⁻² in the numerator and x⁻⁸ in the denominator. When dividing, we subtract the exponents. So, x⁻² / x⁻⁸ becomes x⁻² - (-8) = x⁶.

So far, our A looks like this:

A = (1/50) * (x⁶ * y⁶ * z¹) / (y⁻¹ * z⁻³)

Awesome, we've handled the x part. Now, the expression looks a lot better and less complicated. Understanding how to handle negative exponents is super important. We had to flip some of the negative exponents around to make them positive, but we did it without a problem. When dealing with variables, make sure you're applying the correct rules for division. Remember, when dividing terms with the same base, you subtract the exponents. Keep a close eye on those negative signs – they can be tricky! You might want to write it out on paper for each step. This allows you to keep track of each term and see exactly what's happening. This way, it will be easy for you to double-check your work.

Simplifying the y Variables

Next, let's look at the y variables. We have y⁶ in the numerator and y⁻¹ in the denominator. When dividing, we subtract exponents. So y⁶ / y⁻¹ becomes y⁶ - (-1) = y⁷. We're getting closer! Now, A looks like this:

A = (1/50) * (x⁶ * y⁷ * z¹) / (z⁻³)

See how the y variables are easier to handle. We're getting closer to the final solution. The goal is to have a cleaner, simpler expression, and we're achieving it step by step. Now we have to handle the z variables. This is the final step for variables. If you're struggling, remember the rules: when dividing powers, subtract the exponents. Always remember this, because it is the basis for this entire process. This is why you should practice this as often as possible. This will help you memorize the rules so they become second nature. Now we're on to the home stretch.

Working with the z Variables

Finally, let's tackle the z variables. We have z¹ in the numerator and z⁻³ in the denominator. z¹ / z⁻³ becomes z¹ - (-3) = z⁴. So now our expression A becomes:

A = (1/50) * x⁶ * y⁷ * z⁴

All the exponents are positive! Woohoo! Now, we've reached a form with positive exponents. This is the simplified version we wanted. The expression is much cleaner and easier to understand. This final step is important because it puts everything together. You can see how all the parts of the original expression combined to give us a new, simpler form. Now you have the final result. This is what we wanted. Now you know how to simplify algebraic expressions with positive exponents.

The Final Result: Positive Exponents Achieved!

So, putting it all together, the expression A simplified to: A = (x⁷ * y⁷ * z⁴) / 50

We did it! We took an expression with negative exponents and transformed it into an equivalent expression where all the exponents are positive. The process involved simplifying constants and applying the rules of exponents. Pretty awesome, right? Always remember that the key to simplifying algebraic expressions is to understand and apply the rules of exponents. By following these steps, you can confidently simplify any algebraic expression and rewrite it in a form that's easy to work with. Keep practicing, and you'll become a pro in no time! You guys are awesome, and I hope you enjoyed this lesson! Keep practicing, and you'll become algebra masters in no time. This skill is super useful. So, keep at it, and always remember the basics.

Choosing the Correct Answer

To match our simplified form, we need to look at the options provided. Remember our final answer, (x⁶ * y⁷ * z⁴) / 50. We can see the correct answer from the options is none of them. However, the closest one will be A. But, there is no option with 50 as the denominator. Therefore the answer will be based on the question and the provided answer choices. The result can be a trick question that requires you to carefully examine the options and choose the one that closely matches the correctly simplified expression.

Based on the question and the answer choice, you need to find the closest possible option. The closest one is A, because the exponent of x is 6. Let's remember that we simplified the constant term 2/100 to 1/50. Option A has the numerator as x⁶ * y⁷ * z⁴, and the denominator is 2 * y⁷ * z⁴. However, we know the correct result is (x⁶ * y⁷ * z⁴) / 50. We will pick A as the answer, assuming the problem has a typo or mistake.

Therefore, based on the step-by-step breakdown and the closest option, we pick option A.

Final Answer: A