the square root of 39
The square root of 39 is approximately 6.2449979984.
To find the square root of 39, you can follow these steps:
Step 1: Estimate the square root.
We can start by finding the two perfect square numbers that 39 lies between. The perfect squares closest to 39 are 36 and 49. So, we can estimate that the square root of 39 is between 6 and 7.
Step 2: Use the Babylonian method for approximation.
We will use the Babylonian method to refine our estimate and get a more accurate square root value. The Babylonian method involves iteratively updating the estimate using the following formula:
x_n+1 = (x_n + (39 / x_n)) / 2
Let's start with an initial estimate of x = 6:
x_1 = (6 + (39 / 6)) / 2
x_1 = (6 + 6.5) / 2
x_1 = 12.5 / 2
x_1 = 6.25
Step 3: Repeat the process.
Now we take this new value of x (x_1), and repeat the process to get an even more accurate estimate:
x_2 = (x_1 + (39 / x_1)) / 2
x_2 = (6.25 + (39 / 6.25)) / 2
x_2 = (6.25 + 6.24) / 2
x_2 = 12.49 / 2
x_2 = 6.245
Step 4: Keep repeating until you reach a desired level of accuracy.
You can continue this process for as many times as necessary to get the desired level of accuracy. The more iterations you do, the more accurate your estimate of the square root of 39 will be.
Using this method, you will find that the square root of 39 is approximately 6.245.
To find the square root of 39, you can use a calculator or manual methods such as estimation or long division. Here's an explanation of the manual method:
1. Start by guessing a number that when squared is close to 39. Let's start with the number 6.
6 * 6 = 36
Since 36 is close to 39, we know that the square root of 39 is slightly larger than 6.
2. To refine your estimate, divide 39 by the guess you made (6). This will give you a new estimate.
39 / 6 = 6.5
The new estimate is 6.5.
3. Now, take the average of the guess (6) and the new estimate (6.5).
(6 + 6.5) / 2 = 6.25
The new average is 6.25.
4. Repeat steps 2 and 3 until you reach the desired level of accuracy. With each repetition, the average will get closer and closer to the actual square root of 39.
By using iterative approximation, you can find that the square root of 39 is approximately 6.245.