How to Solve Squeroot Without Calculator Khan Academy

Introduction

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25. While calculators make finding square roots quick and easy, learning how to calculate them manually is a useful skill that can be applied in various situations.

The Khan Academy method is a systematic approach to finding square roots without a calculator. It involves estimating the square root and then refining that estimate through iteration. This method is particularly useful for numbers that don't have perfect square roots, such as 2, 3, 5, 7, etc.

Khan Academy Method

The Khan Academy method involves the following steps:

1. Estimate the square root by finding two perfect squares between which the number lies.
2. Use the average of these two perfect squares as an initial estimate.
3. Square the estimate and compare it to the original number.
4. Adjust the estimate based on whether it's too high or too low.
5. Repeat the process until a sufficiently accurate estimate is obtained.

This method is based on the principle that the square root of a number lies between two consecutive integers. By iteratively refining the estimate, we can approach the true square root with reasonable accuracy.

Step-by-Step Guide

Step 1: Find Two Perfect Squares

Start by identifying two perfect squares between which your number lies. For example, if you're trying to find the square root of 10, you would notice that 3² = 9 and 4² = 16. Since 9 < 10 < 16, the square root of 10 lies between 3 and 4.

Step 2: Calculate the Average

Next, calculate the average of the two perfect squares. For our example, the average of 3 and 4 is (3 + 4) / 2 = 3.5. This is your initial estimate for the square root.

Step 3: Square the Estimate

Square your estimate to see how close it is to the original number. In our example, 3.5² = 12.25. Since 12.25 > 10, our estimate is too high.

Step 4: Adjust the Estimate

If your estimate is too high, take the average of the lower bound and your estimate. If it's too low, take the average of your estimate and the upper bound. In our example, since 3.5 was too high, we take the average of 3 and 3.5, which is 3.25.

Step 5: Repeat the Process

Continue this process, each time squaring your new estimate and adjusting it based on whether it's too high or too low. For our example, the next estimate would be (3 + 3.25) / 2 = 3.125, and so on. After a few iterations, you'll converge on a value close to the actual square root.

Examples

Let's work through a couple of examples to illustrate the method.

Example 1: Square Root of 10

1. Find perfect squares: 3² = 9, 4² = 16 → √10 is between 3 and 4.
2. First estimate: (3 + 4) / 2 = 3.5 → 3.5² = 12.25 (too high).
3. Second estimate: (3 + 3.5) / 2 = 3.25 → 3.25² = 10.5625 (too high).
4. Third estimate: (3 + 3.25) / 2 = 3.125 → 3.125² = 9.7656 (too low).
5. Fourth estimate: (3.125 + 3.25) / 2 = 3.1875 → 3.1875² ≈ 10.16 (close enough).

The actual square root of 10 is approximately 3.1623, so our estimate of 3.1875 is quite close after just four iterations.

Example 2: Square Root of 20

1. Find perfect squares: 4² = 16, 5² = 25 → √20 is between 4 and 5.
2. First estimate: (4 + 5) / 2 = 4.5 → 4.5² = 20.25 (too high).
3. Second estimate: (4 + 4.5) / 2 = 4.25 → 4.25² = 18.0625 (too low).
4. Third estimate: (4.25 + 4.5) / 2 = 4.375 → 4.375² ≈ 19.14 (close enough).

The actual square root of 20 is approximately 4.4721, so our estimate of 4.375 is reasonably accurate after just three iterations.

Limitations

While the Khan Academy method is a useful technique for estimating square roots, it has some limitations:

• It requires multiple iterations to achieve a precise result, which can be time-consuming.
• The method works best for numbers that don't have perfect square roots. For perfect squares, the exact square root can be found more quickly.
• The accuracy of the result depends on the number of iterations performed. More iterations generally lead to more accurate results.

Despite these limitations, the Khan Academy method is a valuable tool for understanding the concept of square roots and for situations where a calculator is not available.