Problem 74 - Prealgebra

Question

Complete the following tasks to estimate the given square root. a) Determine the two integers that the square root lies between. b) Draw a number line, and locate the approximate location of the square root between the two integers found in part (a). c) Without using a calculator, estimate the square root to the nearest tenth. \(\sqrt{12}\)

Step-by-Step Solution

Answer

\(\sqrt{12}\) ≈ 3.5.

Step 1: Identify Perfect Squares

To estimate \( \sqrt{12} \), we first need to identify two consecutive perfect squares that 12 lies between. The nearest perfect squares are 9 and 16, because \( 3^2 = 9 \) and \( 4^2 = 16 \). Therefore, \( \sqrt{12} \) is between 3 and 4.

Step 2: Draw and Locate on Number Line

Draw a number line and mark 3 and 4 on it. To find \( \sqrt{12} \), we need to determine a more precise position between these two integers. Since 12 is closer to 9 than to 16, \( \sqrt{12} \) is closer to 3.5 on the number line.

Step 3: Approximate to the Nearest Tenth

Now, let's estimate \( \sqrt{12} \) more precisely. Calculate a mid-point between 3 and 3.5. We square 3.5 to check its closeness, which gives \( 3.5^2 = 12.25 \), slightly over 12. On the other hand, \( 3.4^2 = 11.56 \), which is a bit under 12. Since 12 is between these, \( \sqrt{12} \) is approximately 3.5 to 3.4. Therefore, rounding to the nearest tenth, \( \sqrt{12} \) is approximately 3.5.

Key Concepts

Understanding the Number Line

The number line is a fundamental mathematical concept that helps visualize the position of numbers relative to each other. It’s simply a line where numbers are placed at equal distances. When estimating square roots, a number line is particularly useful for identifying where the square root falls between two integers. For example, with the calculation of \( \sqrt{12} \), the number line provides a clear visual representation. To use it, you start by marking the two perfect squares that are closest to your number—in this case, 9 and 16—which are perfect squares of 3 and 4, respectively. By highlighting these on a number line, you can easily see that \( \sqrt{12} \) will fall between these two points.With this visual aid, you can focus your attention on locating more precisely where \( \sqrt{12} \) sits between 3 and 4. Since 12 is closer to 9, the number line suggests that \( \sqrt{12} \) is closer to 3.5.

The Role of Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself. Recognizing perfect squares is a crucial skill when estimating square roots because they provide boundary markings that square roots fall between. For instance, to estimate \( \sqrt{12} \), we need to find the nearest perfect squares around the number 12. Calculating the squares of consecutive integers, we find that 9 (since \( 3^2 = 9 \)) and 16 (since \( 4^2 = 16 \)) are the perfect squares closest to 12. Utilizing these gives the bounds that \( \sqrt{12} \) is nestled between the square roots of these perfect squares—namely, 3 and 4. Grasping the significance of perfect squares lets you quickly narrow down the range of your estimation, making the task of finding the square root more manageable.

Nearest Tenth Approximation

To find the square root to the nearest tenth, you further narrow down the interval identified on the number line. You make educated guesses between the consecutive integers, here between 3 and 4, then test these with simple calculations. To zero in on a more precise value for \( \sqrt{12} \), you can check values between 3.0 and 4.0. In this case, squares of 3.4 and 3.5 are considered. Since the square of 3.5 results in 12.25—slightly higher than 12—and the square of 3.4 results in 11.56—slightly lower than 12—this pinpoints \( \sqrt{12} \) close to these two decimals.Since 12 lies between these values but closer to 12.25, rounding \( \sqrt{12} \) to the nearest tenth gives roughly 3.5. Thus, estimating and refining using tests and the number line allows us to make informed approximations.