Problem 28 - Prealgebra

Question

Estimate each square root to the nearest integer. Do not use a calculator. $$\pm \sqrt{396}$$

Step-by-Step Solution

Answer

The square root is approximately \(\pm 20\).

Step 1: Identify perfect squares around 396

To estimate the square root of 396, find two closest perfect squares among whole numbers. The perfect squares are integers squared; around 396, we have 361 (since \(19^2 = 361\)) and 400 (since \(20^2 = 400\)).

Step 2: Determine proximity to these squares

Assess how close 396 is to 361 and 400. Calculate the differences: \(396 - 361 = 35\) and \(400 - 396 = 4\). Notice that 396 is closer to 400.

Step 3: Estimate the square root

Since 396 is closer to 400 than to 361, your estimation is the square root of the number closer to 396, which is 400. Therefore, estimate \(\sqrt{396}\) as being closest to \(\sqrt{400}\), that is 20.

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that are the result of an integer multiplied by itself.
For example, when you multiply 4 by 4, you get 16. That means 16 is a perfect square.
Recognizing perfect squares is crucial in many mathematical problems, especially when estimating square roots.
Some common perfect squares include:

• 1 (since \(1^2 = 1\))
• 4 (since \(2^2 = 4\))
• 9 (since \(3^2 = 9\))
• 16 (since \(4^2 = 16\))
• 25 (since \(5^2 = 25\))

Finding the perfect squares that are closest to a given number helps in estimating its square root.
For the number 396, the closest perfect squares are 361 (\(19^2\)) and 400 (\(20^2\)). This is because they are integers we can easily identify in the nearby range.

Mastering Square Root Estimation

Estimating a square root involves locating the number’s position between two perfect squares.
Once these perfect squares are identified, determine which is closer to the original number.
For example, let's say you need to estimate the square root of 396.
First, find that 396 lies between the perfect squares 361 (\(19^2\)) and 400 (\(20^2\)).
Here are some steps to help:

• Calculate the difference between 396 and 361, which is 35.
• Then, calculate the difference between 400 and 396, which is 4.

Since 396 is much closer to 400 than to 361, the square root of 396 is likely closer to 20 than to 19.
This simple comparison can give a quick estimate without a calculator, emphasizing the importance of understanding nearby perfect squares.

Mastering Integer Approximation

Integer approximation allows us to make educated guesses about numbers that are close to whole numbers.
When dealing with square roots, finding the nearest integer approximation is a key skill.
The process is pretty straightforward. Start by identifying nearby perfect squares.
If we take our example of estimating \(\sqrt{396}\), once we know it falls between 361 and 400:

• Note that 361 corresponds to 19, and 400 corresponds to 20.
• Since 396 is closer to 400, we approximate \(\sqrt{396}\) to the integer 20.

This approach streamlines the estimation process by relying on our understanding of perfect squares and distance comparisons.
Such approximations are not only useful in homework problems but also in real-life situations where a quick mental calculation may be needed.