Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into simplifying algebraic expressions. This can seem a little tricky at first, but trust me, with a systematic approach, you'll be acing these problems in no time. Today, we're going to break down how to simplify the expression $(\frac{2pq^{-2}}{p^3q^2})^{-1}$. This is a classic example that tests your understanding of exponent rules and how to manipulate algebraic fractions. We'll go through it step by step, so even if you're new to this, you'll be able to follow along. So, grab your pencils and let's get started. We'll explore the fundamental principles involved, ensuring you grasp the core concepts necessary for simplifying similar problems in the future. The ability to manipulate and simplify algebraic expressions is a fundamental skill in mathematics, opening doors to solving a wide range of equations and tackling more complex problems. This guide will equip you with the knowledge and techniques to simplify expressions like a pro. This question tests your knowledge of negative exponents and how to apply the power of a quotient rule. By understanding the properties of exponents and how to manipulate algebraic fractions, you can approach this problem strategically and efficiently. Let's make sure you're well-equipped to handle similar algebraic challenges with confidence. We'll start by looking at the rules, then work through the given expression step-by-step to arrive at the solution. Let’s make sure we're on the same page by revisiting the essential rules of exponents.

Understanding the Basics: Exponent Rules

Before we jump into the simplification, let's quickly recap the key exponent rules we'll need. These rules are your best friends in algebra, so make sure you're familiar with them. Knowing these rules is like having the keys to unlock the problem. First, remember the negative exponent rule: $a^{-n} = \frac{1}{a^n}$. This means a term with a negative exponent can be moved to the denominator (or the numerator, if it's in the denominator) and the exponent becomes positive. Secondly, the power of a quotient rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$. This tells us how to handle an expression where a fraction is raised to a power. Each part of the fraction gets raised to that power. Next, the power of a product rule: $(ab)^n = a^n b^n$. When a product is raised to a power, each factor in the product gets raised to that power. Also, the power of a power rule: $(a^m)^n = a^{mn}$. When you have a power raised to another power, you multiply the exponents. Finally, let’s revisit the quotient rule: $\frac{a^m}{a^n} = a^{m-n}$. When dividing terms with the same base, you subtract the exponents. With these rules in mind, we can confidently simplify any expression. These rules are essential tools in your mathematical toolkit and mastering them will allow you to confidently solve various algebraic problems. Understanding the fundamentals is crucial before applying them to more complex situations. Make sure you're comfortable with these rules, because they are going to make our lives a whole lot easier as we solve this problem! Practicing these rules will make you feel confident in dealing with similar problems in the future.

Step-by-Step Simplification of the Expression

Alright, now let's simplify $(\frac{2pq^{-2}}{p^3q^2})^{-1}$ step by step. First, we need to deal with the negative exponent outside the parentheses. Remember, anything raised to the power of -1 is equivalent to its reciprocal. Therefore, we can rewrite the expression as:

$\frac{1}{\frac{2pq^{-2}}{p^3q^2}}$

Now, to simplify this, flip the fraction and rewrite it as:

$\frac{p^3q^2}{2pq^{-2}}$

Next, simplify the variables by applying the quotient rule of exponents. Remember that when dividing terms with the same base, you subtract the exponents. This is where things start to get interesting. Let’s tackle the p terms first. We have $p^3$ in the numerator and p (which is the same as $p^1$) in the denominator. So, $p^3 / p^1$ becomes $p^{3-1} = p^2$. Now, let's handle the q terms. We have $q^2$ in the numerator and $q^{-2}$ in the denominator. Therefore, $q^2 / q^{-2}$ becomes $q^{2-(-2)} = q^{2+2} = q^4$. Now we have: $\frac{p^2q^4}{2}$. The number 2 is in the denominator. So the final result is $\frac{p^2q^4}{2}$. Now, let's see which of the options matches this result. Looking at the answer choices, we're going to compare our simplified form to the options provided. It's really helpful to go back and check your work, making sure you didn't miss any steps or make any calculation errors. Double-checking your work is a good habit to develop. Since our solution does not match any of the given options, we can re-evaluate each step and make sure we performed the operations correctly. Based on the options available, we could rewrite the final result so that the options fit our result. Let's re-evaluate each step and ensure that we performed the operations correctly.

Finding the Correct Answer from the Options

Looking back at our simplified expression, $\frac{p^2q^4}{2}$, we need to find an equivalent form among the answer choices. This is where we need to manipulate the expression to match one of the options. Because our final answer doesn't directly match any of the provided choices, we need to adjust our approach. Let's analyze the options and determine which is equivalent to our result, or how our result can be modified to match them. The answer options involve different forms of the variables and constants, which implies that the final form is going to look a bit different. Let's look at the answer choices.

• A. $(\frac{2}{q^2q})^2$ : This option simplifies to $(\frac{2}{q^3})^2 = \frac{4}{q^6}$. This doesn't match our result.
• B. $(\frac{2}{p^2q^2})^2$: This option simplifies to $\frac{4}{p^4q^4}$. This doesn't match our result.
• C. $(\frac{2}{pq^{-1}})$: This simplifies to $2q/p$. This also doesn't match our result.
• D. $(\frac{2}{pq^{-2}})$: This simplifies to $2q^2/p$. Nope, still not it!
• E. $(\frac{(pq^2)^2}{2})$: This simplifies to $\frac{p^2q^4}{2}$.

So, the answer is E. $(\frac{(pq^2)^2}{2})$. This is the only option that, when simplified, results in $\frac{p^2q^4}{2}$.

Conclusion: Mastering Exponent Rules

Great job, guys! We have successfully simplified the given expression by carefully applying the rules of exponents. Remember, practice is key. The more you work with these rules, the easier they'll become. Always double-check your work and make sure you understand each step. If you're unsure, go back and review the rules. Keep practicing, and you'll become a pro at simplifying algebraic expressions in no time! Keep practicing, and you will develop confidence in simplifying algebraic expressions. Understanding these rules is essential for advanced mathematics, so keep it up!