Simplifying Expressions: Positive Exponents Only!
In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the process of simplifying the expression (x^2 * y) * (-2 * x * y^4 * z2)(-2), focusing specifically on expressing the final answer using only positive exponents. This type of problem often appears in algebra and pre-calculus courses, and mastering it builds a strong foundation for more advanced mathematical concepts. So, let's break down the steps involved and conquer this challenge together!
Understanding the Fundamentals of Exponents
Before we dive into the simplification process, let's refresh our understanding of exponents. An exponent indicates how many times a base is multiplied by itself. For example, x^2 means x multiplied by itself (x * x). Negative exponents represent the reciprocal of the base raised to the positive version of the exponent. That is, x^(-n) is equal to 1 / x^n. This understanding is crucial for dealing with the negative exponent in our given expression. Moreover, when we raise a product to a power, we raise each factor in the product to that power, such as (ab)^n = a^n * b^n. Similarly, when dividing powers with the same base, we subtract the exponents, so x^m / x^n = x^(m-n). Understanding and applying these rules correctly is the key to successfully simplifying complex expressions. Mastering these rules not only helps in simplifying expressions but also in solving equations and understanding various mathematical concepts that rely on exponential relationships. Let’s keep these principles in mind as we proceed with the simplification.
Step-by-Step Simplification Process
Now, let's tackle the simplification of the expression (x^2 * y) * (-2 * x * y^4 * z2)(-2) step-by-step. First, we need to address the negative exponent. Recall that a negative exponent indicates a reciprocal. Therefore, we rewrite the term with the negative exponent as a fraction: (-2 * x * y^4 * z2)(-2) becomes 1 / (-2 * x * y^4 * z2)2. This step is crucial because it transforms the negative exponent into a positive one, which is easier to work with. Next, we apply the power of a product rule, which states that (ab)^n = a^n * b^n. We raise each factor inside the parentheses to the power of 2: 1 / ((-2)^2 * x^2 * (y4)2 * (z2)2). Simplifying further, we get 1 / (4 * x^2 * y^8 * z^4). Remember that when raising a power to a power, we multiply the exponents, hence (y4)2 becomes y^8 and (z2)2 becomes z^4. Now, we have simplified the term with the negative exponent. We can rewrite the original expression by substituting this simplified form.
After dealing with the negative exponent, the expression becomes (x^2 * y) * [1 / (4 * x^2 * y^8 * z^4)]. This sets the stage for the next phase of simplification, where we will combine terms and apply the rules of exponents for division.
Combining Terms and Applying Exponent Rules
Having simplified the term with the negative exponent, we can now combine the terms in the expression. We are dealing with (x^2 * y) * [1 / (4 * x^2 * y^8 * z^4)], which can be rewritten as (x^2 * y) / (4 * x^2 * y^8 * z^4). When dividing terms with the same base, we subtract the exponents. So, for the x terms, we have x^2 / x^2, which simplifies to x^(2-2) = x^0. Remember that any non-zero number raised to the power of 0 is 1, so x^0 = 1. For the y terms, we have y / y^8, which simplifies to y^(1-8) = y^(-7). Since we need to express the answer with positive exponents, we rewrite y^(-7) as 1 / y^7. The z term, z^4, remains in the denominator. Finally, the constant term is 1/4. Putting it all together, we have (1/4) * (1) * (1 / y^7) * (1 / z^4). Therefore, the simplified expression is 1 / (4 * y^7 * z^4). This result showcases the power of understanding and applying exponent rules to effectively simplify algebraic expressions. Always remember to check your work and ensure that all exponents are positive, as requested in the original problem.
Final Result and Conclusion
After meticulously following the steps, the simplified form of the expression (x^2 * y) * (-2 * x * y^4 * z2)(-2), using only positive exponents, is 1 / (4 * y^7 * z^4). This exercise highlights the importance of a strong grasp of exponent rules and the systematic application of these rules to simplify complex expressions. By breaking down the problem into manageable steps, such as dealing with the negative exponent first, then applying the power of a product rule, and finally combining like terms, we can arrive at the correct answer with confidence. Simplification is a crucial skill in mathematics, and problems like this help reinforce the fundamental concepts necessary for success in more advanced topics. Remember, practice makes perfect, so continue to work on similar problems to hone your skills and deepen your understanding of exponents and algebraic simplification. Keep simplifying and exploring the fascinating world of mathematics!
For further exploration of exponent rules and practice problems, you can visit Khan Academy's Algebra Section. This resource offers a wealth of information and exercises to help you master these concepts.
