Simplifying (2w^6x^{-2})^{-3} With Positive Exponents

In this article, we will delve into simplifying algebraic expressions, focusing on the expression (2w^6x^{-2})^{-3}. Our goal is to rewrite this expression using only positive exponents. This involves applying several fundamental rules of exponents, including the power of a product rule, the power of a power rule, and the negative exponent rule. Mastering these rules is crucial for success in algebra and higher-level mathematics. Understanding how to manipulate exponents not only simplifies expressions but also provides a foundation for solving more complex problems in calculus, physics, and engineering. This article aims to provide a clear, step-by-step guide to simplifying the given expression, ensuring that each rule and operation is well-understood. By the end of this discussion, you will be equipped with the knowledge and skills to tackle similar problems with confidence. Let's embark on this mathematical journey to unravel the intricacies of exponents and algebraic simplification. The ability to simplify expressions is a cornerstone of mathematical proficiency, and this article serves as an essential resource for anyone looking to enhance their algebraic skills. We will break down each step in detail, making it easy to follow along and grasp the underlying concepts. So, let’s begin by exploring the initial steps in simplifying (2w^6x^{-2})^{-3}.

Understanding the Rules of Exponents

Before we dive into simplifying the expression (2w^6x^{-2})^{-3}, it's essential to understand the rules of exponents that govern our operations. These rules form the foundation of algebraic manipulation and are critical for simplifying expressions correctly. We'll explore three primary rules: the power of a product rule, the power of a power rule, and the negative exponent rule. Grasping these rules is not just about memorization; it's about understanding how they logically apply in various situations. The power of a product rule states that when a product of terms is raised to a power, each term in the product is raised to that power individually. Mathematically, this is expressed as (ab)^n = a^n b^n. This rule is fundamental in distributing exponents across multiple factors within parentheses. The power of a power rule, perhaps one of the most frequently used rules in simplifying exponents, states that when a power is raised to another power, you multiply the exponents. The formula for this rule is (a^m)^n = a^{mn}. This rule is essential for simplifying expressions where exponents are nested. Lastly, the negative exponent rule is crucial for converting expressions with negative exponents into their positive counterparts. The rule states that a^{-n} = 1/a^n and conversely, 1/a^{-n} = a^n. This rule ensures that we can rewrite expressions using only positive exponents, which is often a requirement in simplified forms. These rules are not just isolated concepts; they work together to allow us to simplify complex expressions. By mastering these rules, you’ll be able to confidently tackle a wide range of algebraic problems. Now, let's apply these rules step-by-step to our expression.

Step 1: Applying the Power of a Product Rule

The first step in simplifying the expression (2w^6x^{-2})^{-3} is to apply the power of a product rule. This rule allows us to distribute the exponent -3 to each term inside the parentheses. Recall that the power of a product rule states that (ab)^n = a^n b^n. In our case, the terms inside the parentheses are 2, w^6, and x^{-2}. Applying the power of -3 to each of these terms, we get: 2^{-3}, (w^6)^{-3}, and (x^{-2})^{-3}. This step is crucial as it breaks down the original complex expression into simpler, more manageable components. By distributing the exponent, we can now focus on simplifying each term individually. This approach not only makes the problem less daunting but also allows us to apply other exponent rules more effectively. It’s important to remember that each term, including the constant 2, is affected by the exponent outside the parentheses. Many students might overlook the constant term, leading to errors in the simplification process. Paying close attention to every term and applying the power of a product rule correctly is vital for achieving the correct answer. Now that we have distributed the exponent, the next step involves simplifying each term using the power of a power rule and the negative exponent rule where applicable. This methodical approach ensures that we handle each component with precision, ultimately leading to a fully simplified expression. Let's move on to the next step and see how we can further simplify these terms.

Step 2: Applying the Power of a Power Rule

Now that we have distributed the exponent -3 to each term inside the parentheses, we move on to the next phase: applying the power of a power rule. This rule is particularly useful when we have an exponent raised to another exponent, as seen in the terms (w^6)^{-3} and (x^{-2})^{-3}. The power of a power rule states that (a^m)^n = a^{mn}, meaning we multiply the exponents. For the term (w^6)^{-3}, we multiply the exponents 6 and -3, resulting in w^{6 * -3} = w^{-18}. Similarly, for the term (x^{-2})^{-3}, we multiply the exponents -2 and -3, which gives us x^{-2 * -3} = x^6. It's essential to pay close attention to the signs of the exponents when multiplying. A common mistake is to overlook the negative sign, which can lead to incorrect results. The multiplication of negative exponents, as in the case of x^{-2 * -3}, results in a positive exponent, which is a crucial detail to remember. By applying the power of a power rule, we have simplified the exponents of w and x. However, we still have w^{-18}, which has a negative exponent. Our goal is to express the final answer using only positive exponents, so we will need to address this in the next step. The application of the power of a power rule is a fundamental step in simplifying exponential expressions, and mastering it is essential for algebraic proficiency. Let's proceed to the next step where we will deal with the negative exponents and finalize the simplification process.

Step 3: Addressing Negative Exponents

After applying the power of a power rule, we have the expression 2^{-3}w^{-18}x^6. The presence of negative exponents in 2^{-3} and w^{-18} indicates that we need to apply the negative exponent rule to express these terms with positive exponents. The negative exponent rule states that a^{-n} = 1/a^n. This means that a term raised to a negative exponent is equivalent to the reciprocal of that term raised to the positive exponent. Applying this rule to 2^{-3}, we get 1/2^3. Similarly, applying it to w^{-18}, we get 1/w^{18}. The term x^6 already has a positive exponent, so it remains unchanged. Now, substituting these back into our expression, we have (1/2^3) * (1/w^{18}) * x^6. This step is critical in fulfilling the requirement of writing the answer using only positive exponents. It's important to remember that moving a term with a negative exponent from the numerator to the denominator (or vice versa) changes the sign of the exponent. This is the essence of the negative exponent rule. Many students find this rule challenging, but with practice, it becomes second nature. By converting the terms with negative exponents to their positive counterparts, we are one step closer to the final simplified form. Now, let's simplify the constant term and combine the terms to arrive at the ultimate answer. This final step will solidify our understanding of the entire simplification process.

Step 4: Final Simplification

Having addressed the negative exponents, our expression now looks like this: (1/2^3) * (1/w^{18}) * x^6. The next step is to simplify the constant term, 1/2^3. We know that 2^3 means 2 * 2 * 2, which equals 8. Therefore, 1/2^3 simplifies to 1/8. Now, substituting this back into our expression, we have (1/8) * (1/w^{18}) * x^6. To combine these terms into a single fraction, we multiply the numerators together and the denominators together. The numerator will be 1 * 1 * x^6 = x^6, and the denominator will be 8 * w^{18} = 8w^{18}. So, the final simplified expression is x^6 / (8w^{18}). This result satisfies the condition of using only positive exponents. This final step demonstrates how all the previous steps come together to achieve the simplified form. It’s a testament to the power of applying the exponent rules systematically and accurately. Simplifying expressions is not just about getting the correct answer; it’s about understanding the process and the underlying mathematical principles. By breaking down the problem into manageable steps, we can tackle even complex expressions with confidence. The ability to simplify expressions like this is a fundamental skill in algebra and is essential for more advanced mathematical studies. Let's recap the entire process to reinforce our understanding.

Conclusion

In conclusion, we have successfully simplified the expression (2w^6x^{-2})^{-3} and rewritten it using only positive exponents. The simplified form is x^6 / (8w^{18}). This process involved several key steps, each utilizing fundamental rules of exponents. We began by applying the power of a product rule to distribute the exponent -3 to each term inside the parentheses. This gave us 2^{-3}(w^6)^{-3}(x^{-2})^{-3}. Next, we applied the power of a power rule to simplify the exponents of w and x, resulting in 2^{-3}w^{-18}x^6. Recognizing the negative exponents, we then applied the negative exponent rule to rewrite 2^{-3} as 1/2^3 and w^{-18} as 1/w^{18}. Finally, we simplified the constant term 1/2^3 to 1/8 and combined all terms to arrive at the final simplified expression x^6 / (8w^{18}). This exercise highlights the importance of understanding and applying the rules of exponents correctly. Each step is crucial, and a mistake in any step can lead to an incorrect final answer. Simplifying expressions is a core skill in algebra and is essential for success in higher-level mathematics. By mastering these techniques, you will be well-equipped to handle more complex problems involving exponents and algebraic manipulation. This article has provided a comprehensive guide to simplifying the given expression, and with practice, you can confidently apply these methods to similar problems. Remember, mathematics is a journey of learning and practice, and each problem solved is a step forward in your mathematical understanding.