Simplifying Exponential Expressions A Comprehensive Guide

In mathematics, simplifying expressions is a fundamental skill. This is especially true when dealing with exponents and algebraic manipulations. In this article, we will delve into a detailed explanation of how to simplify a complex exponential expression. We will start by breaking down the expression step-by-step, explaining the rules of exponents as we go. Our focus will be on the expression ((2a^-3b4)^2 / (3a^5b)-2)^-1, a seemingly intricate problem that can be solved with careful application of exponent rules. We will cover key concepts such as the power of a power rule, the quotient of powers rule, and how to handle negative exponents. This comprehensive guide aims to provide clarity and a thorough understanding, making the simplification process accessible to everyone.

Breaking Down the Problem: Step-by-Step Simplification

To effectively tackle the simplification of the given expression, ((2a^-3b4)^2 / (3a^5b)-2)^-1, we need to systematically apply the rules of exponents. Let's break down each step to ensure clarity and precision. First, we address the outermost exponent, the -1, which essentially means taking the reciprocal of the entire expression. This initial step sets the stage for further simplifications. Then, we will look at how to handle the power of a power, which involves multiplying exponents. This is a crucial rule to remember when dealing with nested exponents. Additionally, we will discuss how to manage negative exponents, which indicate reciprocal values. Understanding how to manipulate these exponents is key to simplifying complex expressions. By the end of this section, you will clearly understand how each rule applies and how it contributes to the final simplified form.

Step 1: Applying the Outer Exponent

The initial expression we are working with is ((2a^-3b4)^2 / (3a^5b)-2)^-1. The first step in simplifying this expression involves addressing the outermost exponent, which is -1. Applying this exponent means taking the reciprocal of the entire fraction inside the parentheses. In other words, we flip the numerator and the denominator. So, when we apply the -1 exponent, the expression transforms from ((2a^-3b4)^2 / (3a^5b)-2)^-1 to ((3a^5b)-2 / (2a^-3b4)^2). This reciprocal operation is a direct application of the rule that states (x/y)^-1 = y/x. Understanding this step is crucial because it reorients the expression, making it easier to work with in subsequent steps. We are essentially redistributing the terms to set up the next phase of simplification, which involves dealing with the inner exponents and coefficients.

Step 2: Power of a Power Rule

After dealing with the outermost exponent, we now focus on the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents: (x^m)n = x^(mn). Applying this to our expression, ((3a^5b)-2 / (2a^-3b4)^2), we first address the numerator, (3a^5b)-2. We distribute the -2 exponent to each term inside the parentheses: 3^-2, a^(5-2), and b^-2. This simplifies to 3^-2 * a^-10 * b^-2. Next, we apply the power of a power rule to the denominator, (2a^-3b4)^2. Distributing the 2, we get 2^2, a^(-32), and b^(42), which simplifies to 2^2 * a^-6 * b^8. So, the entire expression now looks like (3^-2 * a^-10 * b^-2) / (2^2 * a^-6 * b^8). This step is pivotal because it breaks down the complex exponents into simpler terms, paving the way for further simplification by combining like terms.

Step 3: Dealing with Negative Exponents

Continuing our simplification journey, we now tackle the negative exponents in the expression (3^-2 * a^-10 * b^-2) / (2^2 * a^-6 * b^8). A negative exponent indicates that the base should be on the opposite side of the fraction. In other words, x^-n is equivalent to 1/x^n. Applying this rule, we move 3^-2 from the numerator to the denominator, making it 3^2. Similarly, a^-10 moves to the denominator as a^10, and b^-2 moves to the denominator as b^2. In the denominator, a^-6 moves to the numerator as a^6. So, the expression transforms to (a^6) / (2^2 * 3^2 * a^10 * b^2 * b^8). This step is crucial because it converts all negative exponents into positive ones, making the expression easier to manage and combine terms effectively. By understanding how to deal with negative exponents, we are better equipped to simplify complex algebraic expressions.

Step 4: Simplifying and Combining Like Terms

With all the negative exponents addressed, we can now simplify and combine like terms in the expression (a^6) / (2^2 * 3^2 * a^10 * b^2 * b^8). First, let's handle the constants: 2^2 equals 4, and 3^2 equals 9. So, the denominator becomes 4 * 9, which is 36. Next, we combine the 'a' terms. We have a^6 in the numerator and a^10 in the denominator. When dividing like terms with exponents, we subtract the exponents: a^(6-10) which equals a^-4. However, since we want to express our answer with positive exponents, we move a^-4 to the denominator as a^4. For the 'b' terms, we have b^2 and b^8 in the denominator. Multiplying these together, we add the exponents: b^(2+8), which equals b^10. So, the expression simplifies to 1 / (36 * a^4 * b^10). This final step combines all the individual simplifications we've made, presenting a clean and concise form of the original complex expression.

Final Simplified Expression

After meticulously following the steps of simplification, the original expression ((2a^-3b4)^2 / (3a^5b)-2)^-1 is equivalent to 1 / (36a^4b10). This result showcases the power of understanding and applying the rules of exponents. Each step, from addressing the outer exponent to combining like terms, played a crucial role in reducing the complexity of the expression. This final form is much easier to interpret and work with in further mathematical operations. The journey through this simplification process highlights the importance of a systematic approach and a solid grasp of exponent rules in algebraic manipulations.

Conclusion: Mastering Exponential Simplification

In conclusion, the process of simplifying exponential expressions, such as ((2a^-3b4)^2 / (3a^5b)-2)^-1, involves a series of steps that rely on the fundamental rules of exponents. We started by addressing the outer exponent, which involved taking the reciprocal of the expression. We then applied the power of a power rule, distributed exponents, dealt with negative exponents, and finally combined like terms. The result, 1 / (36a^4b10), demonstrates the effectiveness of these rules when applied systematically. Mastering these techniques is crucial for anyone looking to excel in algebra and beyond. By breaking down complex problems into smaller, manageable steps, we can confidently tackle even the most challenging expressions. This article serves as a comprehensive guide, providing a clear and detailed explanation of each step, ensuring that readers can confidently simplify similar expressions in the future.

Choose the Correct Answer

Therefore, the correct answer is:

C. 1 / (36a^4b10)

This detailed walkthrough should provide a comprehensive understanding of how to simplify complex exponential expressions. Remember, practice is key to mastering these concepts!