Simplest Form Of √1,764 A Step-by-Step Solution

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Introduction: Delving into the Realm of Square Roots

In the captivating domain of mathematics, square roots hold a position of paramount importance, serving as fundamental building blocks for a plethora of mathematical concepts and applications. At its core, a square root of a number represents a value that, when multiplied by itself, yields the original number. This fundamental concept has far-reaching implications, permeating various branches of mathematics, including algebra, geometry, and calculus. Understanding square roots is not merely an academic pursuit; it empowers us to solve real-world problems, unravel complex equations, and gain deeper insights into the intricate patterns that govern our universe. From calculating distances and areas to modeling physical phenomena, square roots play an indispensable role in shaping our comprehension of the world around us.

To embark on this mathematical expedition, we set our sights on a specific challenge: determining the simplest form of the square root of 1,764, mathematically expressed as √1,764. This seemingly straightforward problem unveils a treasure trove of mathematical concepts and techniques, inviting us to explore the depths of number theory and algebraic manipulation. By meticulously dissecting the intricacies of this square root, we aim to not only arrive at the solution but also gain a profound appreciation for the underlying principles that govern its behavior. So, let us embark on this journey of mathematical discovery, unraveling the secrets hidden within √1,764 and illuminating the path to its simplest form.

Deconstructing the Problem: Prime Factorization as the Key

To effectively simplify the square root of 1,764, we turn to a powerful technique known as prime factorization. Prime factorization involves breaking down a composite number into its constituent prime numbers, which are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, and so on). This process is akin to dissecting a complex structure into its fundamental building blocks, allowing us to gain a deeper understanding of its composition.

The prime factorization of 1,764 is expressed as 2 × 2 × 3 × 3 × 7 × 7, which can be more concisely written as 2² × 3² × 7². This representation unveils the inherent structure of 1,764, revealing that it is composed of the squares of the prime numbers 2, 3, and 7. This crucial insight forms the cornerstone of our simplification strategy.

Recognizing the prime factors and their exponents is essential because it allows us to rewrite the square root in a more manageable form. By expressing 1,764 as a product of squares, we can leverage the fundamental property of square roots that states √(a × b) = √a × √b. This property allows us to separate the square root of a product into the product of individual square roots, making the simplification process significantly easier.

With the prime factorization of 1,764 at our disposal, we are now poised to embark on the next phase of our journey: simplifying the square root by extracting the perfect squares. This step will bring us closer to unveiling the simplest form of √1,764, revealing its true mathematical essence.

Simplifying the Square Root: Unveiling Perfect Squares

Having meticulously determined the prime factorization of 1,764 as 2² × 3² × 7², we are now equipped to simplify the square root by extracting the perfect squares. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, and so on). In our prime factorization, we observe that 2², 3², and 7² are all perfect squares.

Leveraging the property √(a × b) = √a × √b, we can rewrite √1,764 as √(2² × 3² × 7²) = √2² × √3² × √7². This transformation allows us to isolate the square roots of the perfect squares, which can be readily simplified.

The square root of a perfect square is simply the integer that was squared to obtain the perfect square. Therefore, √2² = 2, √3² = 3, and √7² = 7. Substituting these values back into our expression, we get √1,764 = 2 × 3 × 7.

By extracting the perfect squares from the square root, we have significantly simplified the expression. We have transformed a complex square root into a product of integers, making it much easier to comprehend and manipulate. This step is crucial in our quest to determine the simplest form of √1,764.

In the subsequent section, we will perform the final calculation to arrive at the simplest form of √1,764, revealing the ultimate solution to our mathematical challenge. The meticulous process of prime factorization and perfect square extraction has paved the way for this final step, bringing us closer to unraveling the essence of this square root.

The Final Calculation: Reaching the Simplest Form

With the expression √1,764 simplified to 2 × 3 × 7, we are now on the cusp of determining its simplest form. This final calculation involves a straightforward multiplication of the integers, leading us to the ultimate solution.

Multiplying the integers together, we get 2 × 3 × 7 = 42. Therefore, the simplest form of √1,764 is 42. This result signifies that 42 is the number that, when multiplied by itself, yields 1,764. We have successfully unraveled the square root, transforming it from a complex expression into a concise integer value.

This journey of simplification has showcased the power of prime factorization and perfect square extraction in handling square roots. By breaking down the number into its prime factors, we were able to identify the perfect squares and extract them from the square root, leading us to the simplest form. This process not only provides the solution but also enhances our understanding of the underlying mathematical principles.

In the concluding section, we will summarize our findings and reflect on the significance of this mathematical exploration. The process of simplifying √1,764 has not only provided a numerical answer but has also illuminated the beauty and elegance of mathematical reasoning.

Conclusion: Reflecting on the Solution

In this comprehensive exploration, we embarked on a mathematical journey to determine the simplest form of √1,764. Through the meticulous application of prime factorization and perfect square extraction, we successfully transformed the complex square root into its simplest form: 42.

The process began with understanding the fundamental concept of square roots and recognizing the need for simplification. We then employed prime factorization to break down 1,764 into its constituent prime numbers: 2² × 3² × 7². This prime factorization served as the cornerstone of our simplification strategy, allowing us to identify the perfect squares within the expression.

Next, we leveraged the property √(a × b) = √a × √b to separate the square root of the product into the product of individual square roots. By extracting the perfect squares, we simplified the expression to 2 × 3 × 7.

Finally, we performed the multiplication to arrive at the simplest form: 42. This result signifies that 42 is the number that, when multiplied by itself, yields 1,764. We have successfully simplified the square root, transforming it into a concise integer value.

This mathematical exploration has not only provided a numerical answer but has also highlighted the power and elegance of mathematical reasoning. By employing fundamental principles and techniques, we were able to unravel the complexities of a square root and arrive at its simplest form. The journey has underscored the importance of prime factorization, perfect square extraction, and the properties of square roots in simplifying mathematical expressions.

In conclusion, the simplest form of √1,764 is 42. This solution serves as a testament to the beauty and power of mathematics, demonstrating how fundamental principles can be applied to solve complex problems and gain deeper insights into the world around us.

Therefore, the correct answer is B. 42.

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