Determining The Value Of Square Root Of 75 A Step By Step Solution

Introduction: Delving into the Realm of Square Roots

In the fascinating world of mathematics, square roots hold a significant position, representing the inverse operation of squaring a number. Understanding square roots is crucial for various mathematical concepts, from solving equations to exploring geometric relationships. In this article, we embark on a journey to determine the value of the square root of 75, a seemingly simple yet insightful exploration that will enhance our understanding of numerical estimation and approximation techniques. Our primary focus will be on pinpointing the range within which the value of √75 lies, thereby solidifying our grasp of irrational numbers and their placement on the number line.

The concept of square roots is deeply intertwined with the idea of perfect squares. A perfect square is a number that can be obtained by squaring an integer (a whole number). For instance, 9 is a perfect square because it is the result of 3 multiplied by itself (3² = 9). Conversely, the square root of a perfect square is simply the integer that was squared to produce it. The square root of 9 is 3, denoted as √9 = 3. However, many numbers, like 75, are not perfect squares, which means their square roots are not integers. These numbers fall into the category of irrational numbers, possessing decimal representations that neither terminate nor repeat. Estimating the values of such square roots requires us to employ approximation techniques and leverage our knowledge of perfect squares.

When we encounter a number like 75, which is not a perfect square, we need to employ strategies to estimate its square root. This involves identifying the perfect squares that surround 75. By recognizing that 75 lies between two consecutive perfect squares, we can narrow down the range within which its square root must fall. This process not only gives us an approximate value but also reinforces our understanding of how numbers are positioned relative to their square roots. In the subsequent sections, we will systematically analyze 75 in relation to perfect squares, paving the way for an accurate estimation of its square root.

Identifying Surrounding Perfect Squares: The Key to Estimation

To accurately determine the value of √75, we must first identify the perfect squares that bracket 75. This approach is based on the understanding that the square root function is monotonically increasing; in simpler terms, as the number increases, its square root also increases. Thus, if we can find two perfect squares, one less than 75 and one greater than 75, we can confidently place √75 between the square roots of these perfect squares. This method provides a solid foundation for estimating the value of square roots of non-perfect squares.

Let's begin by listing some perfect squares to facilitate our analysis. We know that 64 is a perfect square (8² = 64), and 81 is also a perfect square (9² = 81). Notice that 75 lies between these two perfect squares: 64 < 75 < 81. This observation is crucial because it allows us to confine the value of √75. Taking the square root of each part of the inequality, we get √64 < √75 < √81. Since we know the square roots of 64 and 81, we can rewrite this as 8 < √75 < 9. This inequality clearly demonstrates that the value of √75 lies between 8 and 9. This initial bracketing is a significant step in our estimation process.

The significance of identifying these surrounding perfect squares cannot be overstated. It allows us to move from the realm of abstract numbers to a concrete interval within which our target value resides. Without this crucial step, estimating √75 would be significantly more challenging. By anchoring 75 between 64 and 81, we transform a seemingly complex problem into a more manageable estimation task. Furthermore, this method is not unique to 75; it can be applied to estimate the square root of any non-perfect square. The principle remains the same: find the nearest perfect squares to establish the bounds.

Estimating the Value: Placing √75 on the Number Line

Having established that √75 lies between 8 and 9, our next task is to refine this estimation and get a more precise sense of its value. While we know it's between 8 and 9, it's beneficial to determine whether it's closer to 8 or closer to 9. This can be achieved by analyzing how close 75 is to the perfect squares of 64 and 81. The closer 75 is to one of these perfect squares, the closer its square root will be to the square root of that perfect square.

We can start by calculating the differences: 75 - 64 = 11, and 81 - 75 = 6. Notice that 75 is only 6 units away from 81, whereas it is 11 units away from 64. This suggests that √75 is closer to √81, which is 9, than it is to √64, which is 8. This comparison is a key step in refining our estimation. It provides a strong indication that the decimal part of √75 will be greater than 0.5, meaning it will be in the latter half of the interval between 8 and 9.

To further refine our estimation, we can try squaring numbers between 8 and 9, such as 8.5. Calculating 8.5² gives us 72.25. This result is less than 75, indicating that √75 is greater than 8.5. We can then try a slightly larger number, such as 8.7. Squaring 8.7 gives us 75.69, which is slightly greater than 75. This means that √75 is between 8.5 and 8.7, and closer to 8.7 than 8.5. This iterative process of squaring values allows us to narrow down the range and get a more accurate estimation. In practice, we can continue this process to achieve even greater precision, but for the purpose of answering the given question, we have sufficiently established that √75 is between 8 and 9.

Conclusion: Determining the Range of √75

In conclusion, through a systematic approach involving the identification of surrounding perfect squares and subsequent refinement using numerical comparisons, we have successfully determined the range within which the value of √75 lies. Our analysis began with recognizing that 75 is situated between the perfect squares 64 and 81, leading us to the crucial understanding that √75 lies between √64 and √81, or equivalently, between 8 and 9. This initial bracketing was a pivotal step in our estimation process.

Furthermore, by comparing the differences between 75 and the surrounding perfect squares, we observed that 75 is closer to 81 than to 64. This observation indicated that √75 is closer to 9 than to 8. We then employed a method of squaring values between 8 and 9 to refine our estimation, which confirmed that √75 is indeed between 8 and 9. This process not only answered the question but also demonstrated a practical technique for estimating the square roots of non-perfect squares. The key takeaway is that by understanding the relationship between numbers and their square roots, and by leveraging our knowledge of perfect squares, we can effectively approximate irrational numbers.

Therefore, based on our comprehensive analysis, we can definitively conclude that the value of √75 lies between 8 and 9. This answer aligns with option C in the original question, showcasing the effectiveness of our estimation approach. This exploration serves as a valuable exercise in numerical estimation and reinforces the importance of understanding the properties of square roots and perfect squares in mathematics. The ability to accurately estimate square roots is a valuable skill that can be applied in various mathematical and real-world scenarios.