Simplifying (3 3/7)² ⋅ (7/24)² A Step-by-Step Guide

Simplify the expression (3 3/7)² ⋅ (7/24)²

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex-looking problems and reduce them to their most basic and understandable forms. This not only makes the expressions easier to work with but also provides a clearer understanding of the underlying mathematical relationships. In this article, we will embark on a journey to simplify the expression (3 3/7)² ⋅ (7/24)². This particular expression involves fractions, mixed numbers, and exponents, making it an excellent example to demonstrate various simplification techniques. Our goal is to break down the expression step by step, explaining each operation and the reasoning behind it. By the end of this article, you will not only be able to simplify this specific expression but also gain a broader understanding of how to approach similar mathematical problems. So, let's dive in and unravel the intricacies of this expression together.

Understanding the Components

Before we begin the simplification process, it's crucial to understand the individual components of the expression (3 3/7)² ⋅ (7/24)². This expression is composed of several key elements, each playing a distinct role in the overall calculation. First, we encounter a mixed number, 3 3/7. A mixed number is a combination of a whole number and a fraction, representing a value greater than one. In this case, 3 3/7 represents three whole units plus three-sevenths of another unit. Mixed numbers are often converted into improper fractions to facilitate mathematical operations. Next, we have the fraction 7/24. A fraction represents a part of a whole, with the numerator (7) indicating the number of parts we have and the denominator (24) indicating the total number of parts the whole is divided into. Understanding fractions is essential for performing operations like multiplication and division. The expression also involves exponents, specifically the exponent ². This exponent signifies that the base (the number or expression inside the parentheses) is multiplied by itself. In our case, both (3 3/7) and (7/24) are squared, meaning they are each multiplied by themselves. Recognizing and understanding these components – mixed numbers, fractions, and exponents – is the first step towards successfully simplifying the expression. Each component requires specific techniques and rules to manipulate, and a clear understanding of these rules is paramount for accurate simplification.

Step 1: Converting the Mixed Number to an Improper Fraction

The initial step in simplifying the expression (3 3/7)² ⋅ (7/24)² is to convert the mixed number, 3 3/7, into an improper fraction. This conversion is necessary because improper fractions are easier to work with in mathematical operations, particularly multiplication and division. To convert a mixed number to an improper fraction, we follow a simple process. We multiply the whole number part (3) by the denominator of the fractional part (7), and then add the numerator of the fractional part (3). This result becomes the new numerator of the improper fraction, while the denominator remains the same. In this case, we multiply 3 by 7, which gives us 21. Then, we add 3 to 21, resulting in 24. Therefore, the improper fraction equivalent of 3 3/7 is 24/7. This conversion is a fundamental skill in fraction manipulation and is crucial for simplifying expressions involving mixed numbers. By converting 3 3/7 to 24/7, we have transformed the expression into a more manageable form, making it easier to proceed with the subsequent steps of simplification. This step highlights the importance of recognizing and applying the appropriate techniques for different types of numbers within an expression. Now that we have an improper fraction, we can move on to the next phase of simplifying the expression.

Step 2: Squaring the Fractions

With the mixed number converted to an improper fraction, our expression now looks like this: (24/7)² ⋅ (7/24)². The next step in simplifying this expression is to square both fractions. Squaring a fraction means multiplying the fraction by itself. This operation involves multiplying the numerators together and the denominators together. For the first fraction, (24/7)², we multiply 24 by 24 to get the new numerator and 7 by 7 to get the new denominator. 24 multiplied by 24 equals 576, and 7 multiplied by 7 equals 49. Therefore, (24/7)² simplifies to 576/49. Similarly, for the second fraction, (7/24)², we multiply 7 by 7, which equals 49, and 24 by 24, which equals 576. Thus, (7/24)² simplifies to 49/576. Squaring fractions is a straightforward process, but it's essential to perform the multiplication accurately to avoid errors in the final result. By squaring both fractions, we have further simplified the expression, making it easier to see the relationship between the two terms. The expression now becomes (576/49) ⋅ (49/576), which sets the stage for the final simplification step. This step underscores the importance of understanding how exponents apply to fractions and how to perform the necessary calculations.

Step 3: Multiplying the Squared Fractions

Having squared both fractions, our expression now stands as (576/49) ⋅ (49/576). The final step in simplifying this expression is to multiply these two fractions together. When multiplying fractions, we multiply the numerators together to get the new numerator and the denominators together to get the new denominator. In this case, we multiply 576 by 49 and 49 by 576. However, before we perform the multiplication, we can observe a crucial detail that will greatly simplify our work. We notice that the numerator of the first fraction (576) is the same as the denominator of the second fraction, and the denominator of the first fraction (49) is the same as the numerator of the second fraction. This means we can simplify the fractions by canceling out these common factors. 576 in the numerator of the first fraction cancels out with 576 in the denominator of the second fraction, and 49 in the denominator of the first fraction cancels out with 49 in the numerator of the second fraction. This cancellation leaves us with 1 in both the numerator and the denominator. Therefore, the result of multiplying (576/49) by (49/576) is 1/1, which is simply equal to 1. This simplification demonstrates the power of recognizing and utilizing cancellation in fraction multiplication. It not only reduces the amount of calculation required but also provides a clear and elegant solution. The final simplified form of the expression (3 3/7)² ⋅ (7/24)² is 1.

Conclusion

In conclusion, we have successfully simplified the expression (3 3/7)² ⋅ (7/24)² by breaking it down into manageable steps. We began by understanding the components of the expression, which included a mixed number, fractions, and exponents. The first step involved converting the mixed number, 3 3/7, into an improper fraction, 24/7. This conversion allowed us to work with fractions more easily. Next, we squared both fractions, (24/7)² and (7/24)², resulting in 576/49 and 49/576, respectively. Finally, we multiplied these squared fractions together and recognized the opportunity to cancel out common factors, leading us to the simplified result of 1. This exercise highlights several important mathematical principles. It demonstrates the importance of converting mixed numbers to improper fractions for ease of calculation. It reinforces the rules of squaring fractions and multiplying fractions. Most importantly, it illustrates the power of simplification through cancellation. By recognizing and utilizing common factors, we can often avoid complex calculations and arrive at the solution more efficiently. The process of simplifying mathematical expressions is not just about finding the answer; it's about developing a deeper understanding of mathematical relationships and problem-solving strategies. By mastering these skills, we can approach more complex problems with confidence and clarity. The simplified form of the expression (3 3/7)² ⋅ (7/24)² is indeed 1, a testament to the elegance and efficiency of mathematical simplification.