Simplifying (27/125)^(2/3) A Step-by-Step Mathematical Guide

In the realm of mathematics, exponents and fractions play a pivotal role in simplifying complex expressions. This article delves into the step-by-step simplification of the expression (27/125)^(2/3), providing a clear and concise explanation suitable for learners of all levels. We will explore the fundamental concepts of exponents and fractions, and how they interact to yield a simplified result. This exploration will not only enhance your understanding of mathematical operations but also equip you with the skills to tackle similar problems with confidence. The key to understanding expressions like these lies in breaking them down into manageable parts and applying the relevant mathematical principles systematically. Let's embark on this mathematical journey together and unravel the intricacies of (27/125)^(2/3).

To effectively simplify (27/125)^(2/3), we first need to understand the structure of the expression. The expression consists of a fraction, 27/125, raised to a fractional exponent, 2/3. This fractional exponent signifies a combination of two operations: a power and a root. The numerator, 2, indicates the power to which the fraction will be raised, while the denominator, 3, indicates the root to be taken. In this case, we are dealing with a cube root because the denominator is 3. To simplify, we can apply the exponent to both the numerator and the denominator separately, turning (27/125)^(2/3) into (27^(2/3)) / (125^(2/3)). This step allows us to handle each part of the fraction independently, making the simplification process more manageable. Recognizing this separation is crucial for tackling more complex expressions involving fractional exponents. By understanding this initial breakdown, we set the stage for further simplification and calculation.

The next crucial step in simplifying (27/125)^(2/3) is to express the numerator and denominator as products of their prime factors. Prime factorization involves breaking down a number into its prime number constituents. This method helps us to identify perfect cubes within the numbers, which is essential for simplifying the cube root. Let's start with the numerator, 27. The prime factorization of 27 is 3 x 3 x 3, which can be written as 3^3. Similarly, for the denominator, 125, the prime factorization is 5 x 5 x 5, which can be written as 5^3. Rewriting the expression using these prime factorizations, we get ((3³⁾(2/3)) / ((5³⁾(2/3)). This transformation is a significant step forward as it allows us to directly apply the properties of exponents and simplify the expression further. Prime factorization is a powerful tool in simplifying expressions involving roots and exponents, and mastering this technique is crucial for success in algebra and beyond. This step clarifies the structure and sets the stage for the final simplification.

Now that we have expressed the numerator and denominator in terms of their prime factors, we can apply the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents. In mathematical terms, (a^m)n = a^(m*n). Applying this rule to our expression, ((3³⁾(2/3)) / ((5³⁾(2/3)), we multiply the exponents in both the numerator and the denominator. For the numerator, 3 is raised to the power of 3, and then the result is raised to the power of 2/3. Multiplying the exponents, we get 3 * (2/3) = 2. So, the numerator simplifies to 3^2. Similarly, for the denominator, we multiply the exponents 3 and 2/3, which again gives us 2. Thus, the denominator simplifies to 5^2. Our expression now becomes (3^2) / (5^2). This simplification significantly reduces the complexity of the expression and brings us closer to the final answer. The power of a power rule is a fundamental concept in algebra, and its application here demonstrates its effectiveness in simplifying complex expressions.

After applying the power of a power rule, our expression has been simplified to (3^2) / (5^2). The final step is to evaluate these powers. 3^2 means 3 multiplied by itself, which equals 9. Similarly, 5^2 means 5 multiplied by itself, which equals 25. Therefore, our expression simplifies to 9/25. This fraction is in its simplest form because 9 and 25 have no common factors other than 1. Thus, the final simplified form of (27/125)^(2/3) is 9/25. This result demonstrates the power of breaking down a complex expression into smaller, manageable parts and applying the relevant mathematical rules systematically. From the initial expression involving fractional exponents and fractions, we have successfully navigated through prime factorization and exponent rules to arrive at a simple, understandable fraction. This process not only provides the solution but also reinforces the importance of understanding the underlying mathematical principles.

In this comprehensive guide, we have successfully simplified the expression (27/125)^(2/3), step by step. Starting with an explanation of the initial expression and the concept of fractional exponents, we moved on to prime factorization, the application of the power of a power rule, and finally, the calculation of the result. The journey through this problem highlights the significance of understanding fundamental mathematical concepts and applying them systematically. Each step, from recognizing the fractional exponent as a combination of a power and a root to utilizing prime factorization and the power of a power rule, played a crucial role in simplifying the expression. The final result, 9/25, not only provides the answer but also underscores the elegance and efficiency of mathematical simplification techniques. This exercise serves as a valuable lesson in problem-solving and reinforces the importance of a strong foundation in mathematical principles. By mastering these techniques, one can approach complex mathematical expressions with confidence and clarity, ultimately enhancing their overall mathematical proficiency.