Problem 73 - 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. \(\sqrt{79}\)

Step-by-Step Solution

Answer

\(\sqrt{79}\) is approximately 8.9.

Step 1: Identify Bounding Integers

To estimate \( \sqrt{79} \), we need two integers between which \( \sqrt{79} \) falls. Calculate the squares of consecutive integers until finding \( 79 \) between them. \( 8^2 = 64 \), \( 9^2 = 81 \). Therefore, \( 8 < \sqrt{79} < 9 \).

Step 2: Draw a Number Line

Draw a number line with major divisions between \( 8 \) and \( 9 \). Mark the points 8, 8.5, and 9 to help us locate \( \sqrt{79} \) on this line. The goal is to visualize that \( \sqrt{79} \) will lie between 8 and 9 closer to 9.

Step 3: Estimate Between Integers

Since \( \sqrt{79} \) is closer to 9 than 8, estimate more accurately by checking the midpoint, which is 8.5. Calculate \( 8.5^2 = 72.25 \). Since \( 79 \) is larger than 72.25, check closer to 9. Let's consider 8.8, for which \( 8.8^2 \approx 77.44 \). Since 79 > 77.44, \( \sqrt{79} \) is slightly larger than 8.8.

Step 4: Final Estimation

Next, try \( 8.9 \) because our previous estimation indicates \( \sqrt{79} > 8.8 \). Calculate \( 8.9^2 \approx 79.21 \). Since 79 is just slightly less than 79.21, the estimate for \( \sqrt{79} \) is \( 8.9 \).

Key Concepts

Bounding Integers

When estimating square roots, the first step is to determine the integers between which the square root lies. This process helps create a boundary for the target square root value. In our specific problem with estimating \( \sqrt{79} \), we need to find two perfect squares that encapsulate 79. Here’s how we do it:

• Identify two consecutive integers whose squares straddle 79—meaning one is less than 79 and the other is more.
• Calculate the squares of several numbers until you find a pair that bounds 79.

For example:

• \( 8^2 = 64 \)
• \( 9^2 = 81 \)

This means that \( \sqrt{79} \) lies between 8 and 9. Recognizing bounding integers creates the foundational step for further refining your square root estimation.

Number Line Estimation

To continue our estimation, we use a number line to better visualize where the square root fits between the bounding integers. A number line provides a visual aid, helping us see the magnitude and specifically locate where \( \sqrt{79} \) will fall between 8 and 9. Here’s how to do it:

• Draw a line segment with endpoints at 8 and 9.
• Mark important intervals, such as 8.0, 8.5, and 9.
• Come up with additional points if necessary for finer accuracy.

By marking points like 8.5 on the number line, we start to notice that \( \sqrt{79} \) appears closer to 9 than to 8. This step helps to refine our estimation even further, moving us closer to an exact approximate value.

Square Root Approximation

Once you have your number line marked and have positioned your estimated square root visually, you can start approximating the square root more precisely. This involves selecting midpoints and checking their squares to narrow down the possibilities.
To estimate \( \sqrt{79} \):

• Check the square of 8.5: \( 8.5^2 = 72.25 \). Since 72.25 is less than 79, the square root must be more than 8.5.
• Try a closer value like 8.8: \( 8.8^2 \approx 77.44 \). Since 77.44 is less than 79, once again, try a higher value.
• Now check 8.9: \( 8.9^2 \approx 79.21 \). Since 79 is slightly less than 79.21, we've found that \( \sqrt{79} \) is approximately 8.9.

By employing this method, students avoid blindly using calculators. Instead, they build a stronger sense of mathematical reasoning and visualization. This practice helps adjust future estimates, providing a balance of visual and calculative intuition.