Verifying Algebraic Simplification Of (6a^2b)/(-2ab^2) Equals (-3a)/b

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In the realm of mathematics, algebraic simplification plays a pivotal role in making complex expressions more manageable and understandable. This article delves into the process of verifying the algebraic simplification of the expression (6a2b)/(-2ab2) to (-3a)/b. We will meticulously dissect each step involved, ensuring a clear and comprehensive understanding of the underlying principles. This detailed exploration will not only confirm the correctness of the simplification but also enhance your proficiency in manipulating algebraic expressions. Grasping these concepts is crucial for success in various mathematical disciplines, making this a valuable exercise for students and enthusiasts alike. By the end of this discussion, you will have a robust understanding of how to simplify algebraic expressions effectively and accurately. So, let's embark on this mathematical journey together, unraveling the intricacies of algebraic simplification.

Breaking Down the Original Expression

Before diving into the simplification process, let's meticulously examine the original expression: (6a2b)/(-2ab2). This fraction involves coefficients, variables, and exponents, all interacting in a specific manner. To effectively simplify this expression, we need to address each component systematically. The numerator, 6a^2b, consists of the coefficient 6 multiplied by the variable 'a' raised to the power of 2 (a^2) and the variable 'b'. The denominator, -2ab^2, comprises the coefficient -2 multiplied by the variable 'a' and the variable 'b' raised to the power of 2 (b^2). Understanding this structure is the first crucial step in our simplification journey. We must recognize the individual elements and their relationships within the expression. This foundational understanding will guide us as we proceed to break down and simplify the expression, ensuring accuracy and clarity in each step. Remember, a thorough grasp of the initial expression is paramount for successful simplification. We will now delve into the process of identifying common factors, a key technique in simplifying algebraic expressions.

Identifying Common Factors

To effectively simplify the expression (6a2b)/(-2ab2), the next crucial step is identifying common factors in both the numerator and the denominator. This process involves examining the coefficients and variables present in both parts of the fraction. Let's start with the coefficients: 6 in the numerator and -2 in the denominator. The greatest common factor (GCF) of 6 and -2 is 2. This means we can divide both coefficients by 2. Next, we analyze the variables. In the numerator, we have a^2 (which is a * a) and b. In the denominator, we have a and b^2 (which is b * b). The common variable factors are 'a' and 'b'. The numerator has 'a' raised to the power of 2, while the denominator has 'a' raised to the power of 1. Thus, 'a' is a common factor. Similarly, both the numerator and denominator have 'b', making it another common factor. Identifying these common factors – 2, a, and b – is essential for the subsequent simplification steps. By recognizing and extracting these shared elements, we pave the way for reducing the expression to its simplest form. This methodical approach ensures that we accurately simplify the expression, maintaining its mathematical integrity throughout the process. Now, let's proceed to the next step, which involves canceling out these identified common factors.

Cancelling Out Common Factors

Now that we've identified the common factors in the expression (6a2b)/(-2ab2), the next step is to cancel them out. This process involves dividing both the numerator and the denominator by the common factors we previously identified: 2, a, and b. Let's begin with the coefficients. Dividing 6 by 2 gives us 3, and dividing -2 by 2 gives us -1. Next, we address the variable 'a'. In the numerator, we have a^2, and in the denominator, we have 'a'. When we divide a^2 by 'a', we are left with 'a' (since a^2 / a = a). Similarly, we have 'b' in the numerator and b^2 in the denominator. Dividing 'b' by b^2 leaves us with 'b' in the denominator (since b / b^2 = 1/b). After canceling out these common factors, we are left with (3a)/(-1b), which can be further simplified. This step is crucial in reducing the expression to its simplest form, making it easier to understand and work with. By systematically canceling out common factors, we ensure that the expression is reduced to its most basic form while maintaining its mathematical equivalence. Now, let's proceed to the final simplification step and see if we arrive at the expected result.

Final Simplification and Verification

After canceling out the common factors, we arrived at the simplified expression (3a)/(-1b). The final step in this process is to tidy up the expression and verify if it matches the target simplification, which is (-3a)/b. To simplify (3a)/(-1b), we can rewrite it by moving the negative sign from the denominator to the front of the fraction or to the numerator. This gives us either -(3a)/(1b) or (-3a)/(b). Since dividing by 1 doesn't change the value, (1b) is simply b. Therefore, the expression becomes (-3a)/b. Comparing this result with the target simplification (-3a)/b, we can clearly see that they match perfectly. This confirms that our simplification process was accurate and that the original expression (6a2b)/(-2ab2) indeed simplifies to (-3a)/b. This verification step is crucial in ensuring the correctness of our work and solidifying our understanding of algebraic simplification. By meticulously working through each step and verifying the final result, we reinforce our skills and confidence in manipulating algebraic expressions. This complete process, from breaking down the original expression to the final verification, provides a comprehensive understanding of algebraic simplification.

Conclusion

In conclusion, the step-by-step process of simplifying the algebraic expression (6a2b)/(-2ab2) has been meticulously demonstrated and verified. We began by breaking down the original expression, identifying its components, and understanding their relationships. Then, we identified the common factors in both the numerator and the denominator, including the coefficients and variables. We systematically canceled out these common factors, reducing the expression to its simplest form. Finally, we performed the final simplification and verified that our result, (-3a)/b, matched the target simplification. This exercise underscores the importance of a systematic approach to algebraic simplification. By carefully breaking down the expression, identifying common factors, canceling them out, and verifying the result, we can confidently simplify complex algebraic expressions. This process not only enhances our mathematical skills but also provides a solid foundation for more advanced mathematical concepts. The ability to simplify algebraic expressions accurately and efficiently is a valuable skill in various fields, making this a fundamental concept to master. This detailed exploration serves as a comprehensive guide to algebraic simplification, empowering you to tackle similar problems with confidence and precision.