Problem 71 - Prealgebra

Question

In Exercises 71-76, 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. \(\sqrt{58}\)

Step-by-Step Solution

Answer

\(\sqrt{58} \approx 7.6\)

Step 1: Determine the Bounding Integers

To find the two integers that \sqrt{58} lies between, identify perfect squares near 58. \(7^2 = 49\) and \(8^2 = 64\). Hence, \(7 < \sqrt{58} < 8\).

Step 2: Draw the Number Line

Place a number line with 7 and 8 on it. Since \(\sqrt{58}\) is greater than 7.5 and closer to 8, place a mark closer to 8 between these integers.

Step 3: Estimate Square Root to the Nearest Tenth

Since 58 is closer to 64 than to 49, \(\sqrt{58}\) is closer to 8 than to 7. By evaluating squares of numbers, find that \(7.6^2 = 57.76\) and \(7.7^2 = 59.29\). \(57.76\) is closer to 58 than 59.29, suggesting \(\sqrt{58} \approx 7.6\).

Key Concepts

Bounding Integers

To estimate a square root like \( \sqrt{58} \), an essential first step is identifying the bounding integers. This involves finding two consecutive whole numbers, "integers," that your square root falls between. In this exercise, we look for perfect squares, because the square root of a perfect square is an integer.
Start by considering perfect squares around 58. Compute squares of whole numbers: \(7^2 = 49\) and \(8^2 = 64\). Notice that 58 is sandwiched between these results. Therefore, the value of \( \sqrt{58} \) must lie between 7 and 8.
The idea of bounding integers sets a foundational step in estimating square roots by narrowing down the range where the root must exist. This approach makes it simpler to guess the square root's approximate value without exactly solving it.

Number Line

Visualizing the position of \( \sqrt{58} \) on a number line is a fantastic way to understand where it sits relative to other numbers. Start drawing a line and marking numbers 7 and 8, which are our bounding integers.
Consider where the number 58 falls between 49 and 64. Since 58 is slightly more than halfway between these two squares, \( \sqrt{58} \) should be nearer to 8 compared to 7. Mark its approximate position closer to the 8 on the number line, further refining our estimate.
This method helps by providing a visual aid that simplifies the idea of approximating a number that isn't neatly a perfect square root, making abstract math more tangible.

Step-by-Step Solution

Breaking down the process of estimating \( \sqrt{58} \) is made easier with a step-by-step approach. Let's go through the steps with clarity:

• Find bounding integers: Identify integers 7 and 8, as \( 7 < \sqrt{58} < 8 \).
• Visualize on number line: Place marks at 7 and 8, and plot \( \sqrt{58} \) closer to 8.
• Estimate to nearest tenth: To sharpen our estimate, compare squares such as \( 7.6^2 = 57.76 \) and \( 7.7^2 = 59.29 \). Notice \( 57.76 \) is closer to 58.

This systematic approach leads to the conclusion that \( \sqrt{58} \approx 7.6 \). By segmenting the process into clear steps, understanding and approximating the square root becomes much simpler, offering a precise estimate without a calculator.