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

This article provides a detailed explanation of how to find the equivalent expression for the given mathematical expression (125^2 / 125^(4/3)). We will explore the fundamental concepts of exponents and fractions, and then apply these principles step-by-step to simplify the expression. This guide aims to help students and anyone interested in mathematics to understand the process clearly and confidently. By the end of this article, you will not only know the correct answer but also understand the underlying mathematical principles that make it so.

Breaking Down the Problem: Exponents and Division

In this mathematical problem, we are tasked with finding an equivalent expression for (125^2 / 125^(4/3)). This involves dealing with exponents and division, two fundamental concepts in algebra. To solve this, we need to understand how exponents work, especially when they are fractions, and how division of terms with the same base can be simplified. Our journey begins with recognizing the base and the exponents involved in the expression.

Identifying the Base and Exponents

The first step in simplifying any exponential expression is to identify the base and the exponents. In our case, the base is 125, and the exponents are 2 and 4/3. The expression 125^2 means 125 multiplied by itself, while 125^(4/3) involves a fractional exponent. Understanding fractional exponents is crucial here because they represent both a power and a root. For example, x^(a/b) is the same as the b-th root of x raised to the power of a. This understanding is key to unlocking the solution.

Understanding Fractional Exponents

Fractional exponents can be intimidating, but they are a powerful tool once understood. An exponent of the form a/b means taking the b-th root of the base and then raising it to the power of a. For instance, x^(2/3) means taking the cube root of x and then squaring the result. In our specific problem, 125^(4/3) means taking the cube root of 125 and then raising it to the power of 4. We'll see how this simplifies our expression in the next steps. The concept of fractional exponents is a bridge connecting powers and roots, which is a recurring theme in advanced mathematics.

Division of Terms with the Same Base

When dividing terms with the same base, a fundamental rule of exponents comes into play: x^m / x^n = x^(m-n). This rule states that you subtract the exponent in the denominator from the exponent in the numerator. Applying this rule is a shortcut that saves us from having to compute the values of large exponents before simplifying. In our problem, we have 125^2 divided by 125^(4/3). According to this rule, we will subtract 4/3 from 2. This step is vital in reducing the expression to a more manageable form, setting the stage for the final simplification.

Step-by-Step Simplification of the Expression

Now, let's simplify the expression (125^2 / 125^(4/3)) step-by-step, applying the principles we've discussed. This methodical approach will clarify each stage of the simplification, ensuring that the final answer is not only correct but also understood in the context of the mathematical rules applied. By following this process, we transform the initial complex expression into a simple, equivalent form.

Applying the Quotient Rule of Exponents

The quotient rule of exponents states that when dividing expressions with the same base, you subtract the exponents. So, (125^2 / 125^(4/3)) becomes 125^(2 - 4/3). The next step is to subtract the fractions in the exponent. This is a crucial step where arithmetic skills come into play, as finding a common denominator is necessary for the subtraction. This rule allows us to combine the numerator and denominator into a single term, significantly simplifying our expression.

Subtracting the Exponents

To subtract the exponents, we need to subtract 4/3 from 2. To do this, we first convert 2 into a fraction with a denominator of 3, which gives us 6/3. Now, the subtraction becomes 6/3 - 4/3, which equals 2/3. Therefore, our expression simplifies to 125^(2/3). This arithmetic operation is a small but significant step, as it transforms the complex fractional subtraction into a simple fraction, making the expression much easier to handle.

Evaluating 125^(2/3)

Now we have 125^(2/3). Remember that a fractional exponent represents both a root and a power. In this case, 2/3 means we need to take the cube root of 125 and then square the result. The cube root of 125 is 5 (since 5 * 5 * 5 = 125). Then, we square 5, which gives us 25. So, 125^(2/3) simplifies to 25. This step is the heart of the simplification process, where we apply the definition of fractional exponents to arrive at a numerical value.

The Final Answer and Its Significance

After simplifying the expression (125^2 / 125^(4/3)), we arrive at the final answer: 25. This answer matches option D. But more than just getting the right answer, it's important to understand what this means in a broader mathematical context. The process we followed demonstrates the power of exponential rules and the importance of understanding fractional exponents. This understanding is crucial for tackling more complex problems in algebra and calculus.

Reviewing the Options

Looking back at the options given:

• A. 1/25
• B. 1/10
• C. 10
• D. 25

We can clearly see that option D, 25, is the correct answer. The other options are incorrect, and understanding why is just as important as getting the right answer. This exercise reinforces the correct application of exponential rules and the importance of careful calculation.

Why Understanding the Process Matters

While arriving at the correct answer is important, truly understanding the process is even more crucial. Mathematics builds upon itself, and the principles used here – understanding exponents, fractional exponents, and division rules – are fundamental to more advanced topics. By mastering these basics, students can approach more complex problems with confidence. The step-by-step simplification we've done is a template for how to approach similar problems, emphasizing the logical progression from one step to the next.

Further Practice and Resources

To solidify your understanding of exponents and fractional exponents, it's beneficial to practice with similar problems. Seek out additional resources such as textbooks, online tutorials, and practice worksheets. Working through a variety of problems will help you internalize the rules and develop your problem-solving skills. Mathematics is a skill that improves with practice, and the more you engage with these concepts, the more comfortable and confident you will become.

Additional Practice Problems

Try simplifying the following expressions:

• (64^2 / 64^(1/3))
• (81^(3/4) / 81^(1/2))
• (32^(4/5) / 32)

Working through these problems will reinforce your understanding of the concepts we've discussed and help you develop your problem-solving skills.

Online Resources and Further Learning

Numerous online resources can help you further explore exponents and other mathematical topics. Websites like Khan Academy, Coursera, and edX offer courses and tutorials on various math subjects. Additionally, many textbooks and workbooks provide comprehensive explanations and practice problems. Taking advantage of these resources can significantly enhance your understanding and skills in mathematics.

Conclusion: Mastering Exponents for Mathematical Success

In conclusion, simplifying expressions with exponents and fractional exponents is a critical skill in mathematics. By understanding the fundamental rules and practicing regularly, you can confidently tackle these types of problems. Our detailed walkthrough of the expression (125^2 / 125^(4/3)) provides a clear example of how to approach such problems step-by-step. Remember, the key is to break down complex problems into smaller, manageable steps, and to understand the underlying principles. This approach not only leads to the correct answer but also builds a solid foundation for future mathematical challenges. This mastery of exponents is a stepping stone to success in algebra and beyond, empowering students to tackle more complex equations and concepts.