Simplify (125^2 / 125^(4/3)) A Step-by-Step Guide

In the realm of mathematics, particularly in algebra and precalculus, simplifying and understanding exponential expressions is a fundamental skill. These expressions, often appearing complex at first glance, can be elegantly resolved by applying the basic rules and properties of exponents. In this comprehensive guide, we will delve into the process of simplifying the expression (125^2 / 125^(4/3)), providing a step-by-step breakdown that elucidates the underlying principles. Our aim is not just to arrive at the correct answer but also to equip you with a robust understanding of how to tackle similar problems effectively. The essence of simplifying such expressions lies in recognizing common bases and leveraging the power of exponent rules, which allow us to manipulate and reduce expressions to their simplest forms. This exploration will involve converting numbers to their prime factorizations, applying exponent rules for division, and finally, arriving at a solution that reflects a clear, concise simplification of the initial expression. By the end of this discussion, you should feel confident in your ability to handle exponential expressions and appreciate the logical progression required to solve them.

To effectively address the problem (125^2 / 125^(4/3)), our initial step involves dissecting the expression into its core components. This methodical approach is crucial for transforming what may seem like a daunting task into a manageable sequence of steps. The first element we pinpoint is the base, which in this case is the number 125. Recognizing the base is pivotal because it allows us to identify potential simplifications through prime factorization. The number 125 is not a prime number, but it is a power of 5, specifically 5 cubed (5^3). This realization is a cornerstone of our simplification strategy because it enables us to rewrite the entire expression in terms of the base 5. Furthermore, we observe that the expression involves exponents, both in the numerator and the denominator. The numerator has an exponent of 2, while the denominator has an exponent of 4/3. These exponents dictate how the base is multiplied by itself, and they are central to the rules we will apply for simplification. Understanding these exponent values is critical because the properties of exponents govern how we combine and reduce expressions. By meticulously examining the base and the exponents, we lay the groundwork for applying the quotient rule of exponents, which will be instrumental in streamlining the expression. This preliminary analysis transforms the problem from an abstract challenge into a concrete task of applying well-defined mathematical principles.

At the heart of simplifying the expression (125^2 / 125^(4/3)) lies the strategic application of prime factorization. This technique involves breaking down the number 125 into its prime factors, which in this case is solely the number 5. The recognition that 125 is equivalent to 5^3 (5 cubed) is a critical juncture in our simplification process. By substituting 125 with 5^3, we fundamentally change the structure of the expression, making it more amenable to the application of exponent rules. This substitution transforms the original expression into ((5^3)^2 / (5^3)^(4/3)). The beauty of this conversion lies in the fact that it allows us to leverage the power of exponents more effectively. We are now dealing with a base that is a prime number, which simplifies the subsequent steps considerably. The exponents, which were initially applied to 125, are now applied to 5^3, opening the door for the application of the power of a power rule. This rule, a cornerstone of exponent manipulation, states that when you raise a power to another power, you multiply the exponents. By making this crucial substitution, we have not only simplified the base but also set the stage for a more straightforward application of exponent rules, ultimately paving the way for a clear and concise solution.

With the base now expressed in its prime form, (5^3), we can effectively apply the exponent rules to simplify the expression ((5^3)^2 / (5^3)^(4/3)). The first rule we invoke is the power of a power rule, which dictates that when you raise a power to another power, you multiply the exponents. Applying this rule to the numerator, (5^3)^2, we multiply the exponents 3 and 2, resulting in 5^6. Similarly, in the denominator, (5^3)^(4/3), we multiply the exponents 3 and 4/3. The multiplication of 3 by 4/3 yields 4, so the denominator simplifies to 5^4. Our expression now takes a simplified form: (5^6 / 5^4). The next pivotal rule we employ is the quotient rule of exponents. This rule states that when dividing expressions with the same base, you subtract the exponents. In our case, we have (5^6 / 5^4), which simplifies to 5^(6-4). Subtracting the exponents 6 and 4 gives us 2, resulting in the simplified expression 5^2. These exponent rules serve as powerful tools, allowing us to condense complex expressions into more manageable forms. By sequentially applying the power of a power rule and the quotient rule, we have methodically reduced the initial expression to a much simpler representation, setting the stage for the final calculation.

Having diligently applied the prime factorization and exponent rules, we've arrived at the simplified expression 5^2. The final step in our journey is to evaluate this expression to obtain the numerical answer. The expression 5^2 signifies 5 raised to the power of 2, which is a straightforward calculation: 5 multiplied by itself. This operation, 5 * 5, yields the result 25. Therefore, the simplified value of the original expression (125^2 / 125^(4/3)) is 25. This result neatly encapsulates the series of transformations we've undertaken, from prime factorization to the application of exponent rules. It underscores the power of these mathematical tools in simplifying complex expressions to their fundamental values. The answer of 25 represents the culmination of our efforts, providing a clear and concise solution to the initial problem. It also serves as a testament to the systematic approach we adopted, which not only led us to the correct answer but also provided a deeper understanding of the underlying mathematical principles.