Simplify The Expression (125^2 / 125^(4/3)) A Step By Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. This article delves into the process of finding an equivalent expression for the given mathematical expression: (125^2 / 125^(4/3)). We will break down the steps involved, apply relevant exponent rules, and arrive at the correct answer. This exploration will not only provide the solution but also enhance understanding of exponent manipulations.

Understanding the Problem

The problem presents us with a fraction where both the numerator and the denominator involve powers of the same base, 125. The numerator is 125 raised to the power of 2, while the denominator is 125 raised to the power of 4/3. To simplify this expression, we need to apply the rules of exponents, specifically the quotient rule, which states that when dividing exponents with the same base, you subtract the powers.

Before we dive into the calculations, it's crucial to remember the key exponent rules. These rules are the foundation for simplifying expressions involving powers. Here's a brief recap:

  • Product Rule: a^m * a^n = a^(m+n)
  • Quotient Rule: a^m / a^n = a^(m-n)
  • Power Rule: (am)n = a^(m*n)
  • Zero Exponent Rule: a^0 = 1
  • Negative Exponent Rule: a^(-n) = 1/a^n
  • Fractional Exponent Rule: a^(m/n) = nth root of (a^m)

Understanding and applying these rules correctly is essential for simplifying complex expressions. In this particular problem, the quotient rule and the fractional exponent rule will be most relevant.

Step-by-Step Solution

1. Applying the Quotient Rule

Our expression is (125^2 / 125^(4/3)). According to the quotient rule, we subtract the exponents:

125^(2 - 4/3)

To subtract the fractions, we need a common denominator, which is 3. So, we rewrite 2 as 6/3:

125^(6/3 - 4/3)

Now we can subtract:

125^(2/3)

2. Interpreting the Fractional Exponent

The expression 125^(2/3) involves a fractional exponent. A fractional exponent indicates both a power and a root. The numerator (2) represents the power to which the base is raised, and the denominator (3) represents the index of the root to be taken. In this case, 125^(2/3) means the cube root of 125 squared.

We can express this as:

(∛125)^2

3. Evaluating the Cube Root

We know that the cube root of 125 is 5 because 5 * 5 * 5 = 125. Therefore:

∛125 = 5

4. Squaring the Result

Now we need to square the result from the previous step:

5^2 = 25

Therefore, the equivalent expression for (125^2 / 125^(4/3)) is 25.

Analyzing the Answer Choices

We have arrived at the solution, which is 25. Now let's look at the given answer choices:

A. 1/25 B. 1/10 C. 10 D. 25

Our calculated answer, 25, matches option D. Therefore, option D is the correct answer.

Common Mistakes and How to Avoid Them

When dealing with exponents and fractional powers, there are several common mistakes students often make. Recognizing these pitfalls can help you avoid them.

  1. Incorrectly Applying the Quotient Rule: A common mistake is to divide the bases instead of subtracting the exponents. Remember, the quotient rule applies only when dividing exponents with the same base.
  2. Misinterpreting Fractional Exponents: Students may confuse the numerator and denominator in a fractional exponent, leading to incorrect calculations of roots and powers. Always remember that the denominator indicates the root, and the numerator indicates the power.
  3. Order of Operations: Forgetting the correct order of operations (PEMDAS/BODMAS) can lead to errors. In this case, it's important to evaluate the exponent before performing any other operations.
  4. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect subtraction or multiplication, can lead to a wrong final answer. Double-checking calculations is always a good practice.

To avoid these mistakes, it is essential to practice regularly and understand the underlying concepts rather than memorizing formulas. Taking the time to write out each step can also help catch errors early on.

Alternative Approaches

While we solved the problem using the quotient rule and fractional exponents, there's an alternative approach that can be helpful, especially for those who prefer working with whole numbers. This approach involves expressing the base (125) as a power of its prime factor (5).

1. Expressing 125 as a Power of 5

We know that 125 is 5 cubed (5^3). We can substitute this into our original expression:

(125^2 / 125^(4/3)) = ((53)2 / (53)(4/3))

2. Applying the Power Rule

Now we apply the power rule, which states that (am)n = a^(m*n):

((53)2 / (53)(4/3)) = (5^(32) / 5^(3(4/3)))

This simplifies to:

(5^6 / 5^4)

3. Applying the Quotient Rule

Now we can apply the quotient rule:

5^(6-4) = 5^2

4. Evaluating the Result

Finally, we evaluate 5^2:

5^2 = 25

This alternative method arrives at the same answer, 25, but it demonstrates how different approaches can be used to simplify the same expression. Choosing the method that best suits your understanding and problem-solving style can make complex problems more manageable.

Real-World Applications of Exponents

Exponents are not just abstract mathematical concepts; they have numerous real-world applications. Understanding exponents can help us in various fields, including:

  1. Science: In physics, exponents are used to express very large or very small numbers, such as the speed of light or the mass of an electron. In chemistry, they are used in calculations involving pH levels and reaction rates. In biology, exponents are used to model population growth and decay.
  2. Computer Science: Exponents are fundamental in computer science, especially in binary code and data storage. Computer memory is often measured in powers of 2 (e.g., kilobytes, megabytes, gigabytes).
  3. Finance: Compound interest calculations heavily rely on exponents. Understanding exponential growth is crucial for making informed financial decisions.
  4. Engineering: Engineers use exponents in various calculations, such as determining the strength of materials, designing circuits, and analyzing signal processing.
  5. Everyday Life: Exponents can be seen in everyday situations, such as understanding how quickly a viral video spreads or calculating the total cost of an item with sales tax.

By understanding the power of exponents, we can better grasp the world around us and solve real-world problems more effectively.

Conclusion

In this article, we successfully simplified the expression (125^2 / 125^(4/3)) and found its equivalent form to be 25. We achieved this by applying the quotient rule and the fractional exponent rule. We also explored an alternative approach using prime factorization and discussed common mistakes to avoid. Furthermore, we highlighted the real-world applications of exponents, emphasizing their importance in various fields.

Mastering exponent rules is essential for success in mathematics and related disciplines. By practicing regularly and understanding the underlying concepts, you can confidently tackle complex expressions and apply them to real-world problems. Remember, mathematics is not just about finding the right answer; it's about developing critical thinking and problem-solving skills that can benefit you in all aspects of life.

The expression (125^2 / 125^(4/3)) presents an interesting problem in simplifying exponents. This mathematical problem falls under the category of algebra, specifically dealing with exponent rules and simplification. The core task is to find an equivalent expression to the given fraction. This involves understanding how exponents behave when dividing numbers with the same base and how fractional exponents work. The ability to simplify such expressions is crucial not only in academic mathematics but also in various fields such as physics, engineering, and computer science, where exponential notations are frequently used to represent very large or very small quantities. By working through this problem, one can solidify their understanding of exponent manipulation and gain confidence in handling similar mathematical challenges.

Understanding the Problem Statement

Before diving into the solution, let's dissect the problem statement to ensure we fully grasp what's being asked. We are given an expression, (125^2 / 125^(4/3)), and our objective is to find an equivalent expression from the provided options. This means we need to simplify the given expression using the rules of exponents until we arrive at a form that matches one of the choices. The problem involves a fraction where both the numerator and the denominator have the same base (125) raised to different powers. This immediately suggests that the quotient rule of exponents will be a key tool in simplifying the expression. Additionally, the presence of a fractional exponent (4/3) indicates that we will need to understand how to interpret and manipulate fractional powers, which link exponents to roots and powers. By carefully examining the structure of the expression, we can develop a strategic approach to solving the problem efficiently.

Key Concepts Involved

To effectively tackle the problem of simplifying (125^2 / 125^(4/3)), a strong understanding of several key concepts is required. Firstly, the quotient rule of exponents is paramount. This rule states that when dividing two powers with the same base, you subtract the exponents: a^m / a^n = a^(m-n). This will allow us to combine the terms in the numerator and denominator into a single expression with 125 as the base. Secondly, understanding fractional exponents is crucial. A fractional exponent, such as 4/3, represents both a power and a root. The numerator (4) is the power to which the base is raised, and the denominator (3) is the index of the root to be taken. Therefore, 125^(4/3) can be interpreted as the cube root of 125 raised to the power of 4. Lastly, being familiar with perfect cubes and perfect squares is beneficial. Recognizing that 125 is a perfect cube (5^3) will help simplify the expression further. By mastering these concepts, one can confidently navigate the problem-solving process and arrive at the correct solution.

Step-by-Step Solution to the Problem

Now, let's proceed with a detailed, step-by-step solution to find the equivalent expression for (125^2 / 125^(4/3)). This methodical approach will not only provide the answer but also enhance understanding of the simplification process.

Step 1: Applying the Quotient Rule

The first step involves applying the quotient rule of exponents. As we established, this rule states that a^m / a^n = a^(m-n). In our expression, 125 is the common base, and the exponents are 2 and 4/3. Applying the quotient rule, we subtract the exponents:

125^(2 - 4/3)

This step combines the numerator and denominator into a single term, making the expression easier to handle.

Step 2: Simplifying the Exponent

Next, we need to simplify the exponent (2 - 4/3). To subtract these numbers, we need a common denominator, which is 3. We can rewrite 2 as 6/3, so the expression becomes:

125^(6/3 - 4/3)

Now we can subtract the fractions:

125^(2/3)

This step simplifies the exponent to a single fraction, setting the stage for further simplification using the understanding of fractional exponents.

Step 3: Interpreting the Fractional Exponent

Here, we interpret the fractional exponent (2/3). As discussed earlier, a fractional exponent represents both a root and a power. The denominator (3) indicates the cube root, and the numerator (2) indicates the power to which the base is raised. Thus, 125^(2/3) can be expressed as:

(∛125)^2

This step transforms the fractional exponent into a more understandable form, making it easier to evaluate.

Step 4: Evaluating the Cube Root

Now, we evaluate the cube root of 125. We know that 125 is a perfect cube, specifically 5^3 (5 * 5 * 5 = 125). Therefore, the cube root of 125 is 5:

∛125 = 5

Substituting this back into our expression, we get:

5^2

This step simplifies the root, bringing us closer to the final answer.

Step 5: Squaring the Result

The final step is to square the result from the previous step. We have 5^2, which means 5 multiplied by itself:

5^2 = 5 * 5 = 25

Therefore, the equivalent expression for (125^2 / 125^(4/3)) is 25. This completes the simplification process, giving us the final answer.

Identifying the Correct Answer Choice

After performing the step-by-step simplification, we arrived at the result of 25. Now, let's consider the given answer choices to identify the correct option.

The provided answer choices are:

A. 1/25 B. 1/10 C. 10 D. 25

Comparing our result (25) with the given options, it is clear that option D (25) matches our simplified expression. Therefore, option D is the correct answer. This confirms that our methodical approach has led us to the accurate solution.

Common Mistakes and How to Avoid Them

When simplifying expressions with exponents, it's easy to make mistakes if you're not careful. Understanding these common errors can help you avoid them and ensure accuracy in your calculations. Here are some pitfalls to watch out for:

  1. Misapplying the Quotient Rule: A frequent mistake is to divide the bases instead of subtracting the exponents when using the quotient rule (a^m / a^n = a^(m-n)). Always remember that the bases should remain the same, and only the exponents are subtracted.
  2. Incorrectly Handling Fractional Exponents: Fractional exponents can be tricky. A common error is to mix up the numerator and denominator, leading to incorrect calculation of roots and powers. Always remember that the denominator is the index of the root, and the numerator is the power.
  3. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect subtraction or multiplication, can derail the entire solution. Double-checking your calculations, especially when dealing with fractions, is crucial.
  4. Forgetting the Order of Operations: Failing to follow the correct order of operations (PEMDAS/BODMAS) can lead to errors. Ensure you address exponents before other operations like multiplication or division.

Strategies to Avoid Mistakes

To minimize errors when simplifying expressions with exponents, consider the following strategies:

  • Write Out Each Step: Clearly writing out each step in the simplification process helps you keep track of your work and identify potential errors more easily.
  • Double-Check Calculations: Take the time to double-check your arithmetic, especially when subtracting or multiplying fractions. A small mistake can lead to a wrong final answer.
  • Understand the Rules: Don't just memorize the rules of exponents; understand why they work. This deeper understanding will help you apply them correctly in various situations.
  • Practice Regularly: The more you practice simplifying expressions, the more comfortable and confident you'll become. Regular practice also helps reinforce the rules and identify patterns.

By being aware of common mistakes and adopting these strategies, you can enhance your accuracy and problem-solving skills in simplifying expressions with exponents.

Alternative Approaches to Solving the Problem

While we've solved the problem using the quotient rule and fractional exponents, exploring alternative approaches can provide additional insights and problem-solving skills. Here's an alternative method to simplify the expression (125^2 / 125^(4/3)):

Alternative Method: Prime Factorization

This approach involves expressing the base (125) as a power of its prime factor (5). Since 125 is 5 cubed (5^3), we can substitute this into the original expression:

(125^2 / 125^(4/3)) = ((53)2 / (53)(4/3))

Now, we apply the power rule of exponents, which states that (am)n = a^(m*n). Applying this rule to both the numerator and denominator, we get:

((53)2 / (53)(4/3)) = (5^(32) / 5^(3(4/3)))

Simplifying the exponents:

(5^6 / 5^4)

Now, we can apply the quotient rule, which states that a^m / a^n = a^(m-n):

5^(6-4) = 5^2

Finally, we evaluate 5^2:

5^2 = 25

This alternative method, using prime factorization, arrives at the same answer of 25. It demonstrates that there can be multiple paths to the solution, and choosing the approach that best suits your understanding and problem-solving style can be beneficial. By understanding different methods, you can develop a more flexible and comprehensive approach to mathematical problem-solving.

Real-World Applications of Exponents and Simplification

Exponents and simplification techniques are not just theoretical mathematical concepts; they have significant real-world applications across various fields. Understanding these applications can underscore the importance of mastering these skills.

  1. Science and Engineering:

    • In physics, exponents are used extensively to represent very large and very small numbers, such as the speed of light (approximately 3 x 10^8 meters per second) or the mass of an electron (approximately 9.11 x 10^-31 kilograms). Simplifying expressions with exponents is crucial in scientific calculations.
    • Engineers use exponential functions to model growth and decay phenomena, such as the charging and discharging of capacitors in electrical circuits or the decay of radioactive materials. Simplifying these exponential expressions is vital for accurate modeling and predictions.

  2. Computer Science:

    • In computer science, exponents are fundamental in understanding data storage and processing. Computer memory is measured in powers of 2 (e.g., kilobytes, megabytes, gigabytes), and file sizes are often represented using exponential notation.
    • Algorithms involving exponential time complexity (e.g., O(2^n)) are a key concept in computer science. Understanding how to simplify and analyze these expressions is crucial for algorithm design and optimization.

  3. Finance:

    • Compound interest calculations heavily rely on exponential functions. The formula for compound interest (A = P(1 + r/n)^(nt)) involves raising a quantity to an exponent. Simplifying these expressions is essential for financial planning and investment analysis.
    • Exponential growth models are used to represent the growth of investments, populations, and other quantities. Simplifying these models helps in making predictions and informed decisions.

  4. Everyday Life:

    • Exponents are used in scales like the Richter scale for measuring earthquake intensity or the decibel scale for measuring sound intensity. Understanding exponential scales helps in interpreting and comparing these measurements.
    • Viral phenomena, such as the spread of information on social media or the spread of a disease, often follow exponential growth patterns. Simplifying exponential expressions can provide insights into these processes.

By recognizing the wide-ranging applications of exponents and simplification, we can appreciate the practical value of these mathematical skills and their impact on various aspects of our lives.

Conclusion and Key Takeaways

In this comprehensive exploration, we successfully simplified the expression (125^2 / 125^(4/3)) and found its equivalent form to be 25. We achieved this by applying the quotient rule of exponents, understanding fractional exponents, and simplifying the resulting expression step by step. Additionally, we explored an alternative method using prime factorization, demonstrating that multiple approaches can lead to the same solution.

Key Takeaways from this Problem:

  1. Quotient Rule of Exponents: Mastering the quotient rule (a^m / a^n = a^(m-n)) is crucial for simplifying expressions involving division of powers with the same base.
  2. Fractional Exponents: Understanding how to interpret and manipulate fractional exponents is essential. A fractional exponent represents both a root and a power.
  3. Step-by-Step Simplification: Breaking down complex expressions into smaller, manageable steps enhances accuracy and understanding.
  4. Alternative Methods: Exploring alternative methods can provide additional insights and problem-solving skills.
  5. Real-World Applications: Recognizing the real-world applications of exponents underscores their practical value in various fields.

By mastering these key concepts and techniques, you can confidently tackle similar problems involving exponents and simplification. Continuous practice and a solid understanding of the underlying principles will enhance your mathematical skills and problem-solving abilities.