Problem 27 - Prealgebra

Question

Estimate each square root to the nearest integer. Do not use a calculator. $$\pm \sqrt{200}$$

Step-by-Step Solution

Answer

The nearest integers for \(\pm \sqrt{200}\) are \(\pm 14\).

Step 1: Identify Perfect Squares

First, identify the perfect squares nearest to 200. We know that \(14^2 = 196\) and \(15^2 = 225\). The square root of 200 will be between these two integers, specifically between 14 and 15.

Step 2: Approximate the Square Root

Since \(196\) is closer to \(200\) than \(225\) and \(200\) is closer to \(14\) than to \(15\), \(\sqrt{200}\) is closer to 14.

Step 3: Determine Nearest Integer

Given that \(14^2 = 196\) is only 4 units away from \(200\) and \(15^2 = 225\) is 25 units away, it's clear that 14 is the nearest integer approximation.

Step 4: Consider the Negative Square Root

Since the exercise asks for both positive and negative square roots, the estimates are \(+\sqrt{200} \approx +14\) and \(-\sqrt{200} \approx -14\).

Key Concepts

Perfect Squares

Perfect squares are numbers that are the square of an integer. They offer a quick way to make estimations because they provide reference points.
For example, perfect squares like 1, 4, 9, 16, and so on (up to 15 squared which is 225) are numerical benchmarks against which we can measure the proximity of other numbers.

• A perfect square is calculated as the product of an integer multiplied by itself. For instance, \( 14^2 = 196 \), making 196 a perfect square.
• In the exercise, our task is to find nearby perfect squares to 200, such as 196 and 225.

These surrounding perfect squares help in narrowing down where the square root of a non-perfect square number lies.

Integer Approximation

When estimating the square root of a number like 200, we look for the closest integer values. This process is called integer approximation.
By identifying the perfect squares around 200, we first determine that it lies between two integers, 14 and 15.

• Since \(196\) (\(14^2\)) is closer to 200 than \(225\) (\(15^2\)), \(\sqrt{200}\) is estimated to be closer to 14.
• The closer perfect square allows us to approximate \(\sqrt{200}\) to the nearest integer, which in this case is 14.

This integer approximation is a useful method when calculators can't be used, offering a practical estimate based on spatial reasoning between perfect squares.

Positive and Negative Square Roots

Square roots aren't limited to just positive solutions; they can also have negative counterparts. When asked to estimate \(\pm \sqrt{200}\), it's important to provide both values.

• The symbol \(\pm\) signifies both the positive and negative roots.
• For the positive square root estimation of 200, we have \(+\sqrt{200} \approx +14\), based on our earlier approximation.
• For the negative square root, we mirror this estimate as \(-\sqrt{200} \approx -14\).

This concept illustrates that every positive number has two square roots: one positive and one negative. Each serves different mathematical contexts but is essential for complete solutions.