How to Find Square Root of Non Perfect Squares
Finding the square root of a non-perfect square can sometimes be challenging, especially if you’re not familiar with mathematical techniques. However, there are several methods you can use to approximate the square root of any given number. In this article, we will discuss some of the most common methods to find the square root of non-perfect squares.
1. Estimation Method
One of the simplest ways to find the square root of a non-perfect square is by using the estimation method. This involves finding two perfect squares that are closest to the given number and then approximating the square root by taking the average of the square roots of these two perfect squares.
For example, to find the square root of 20, you would look for the perfect squares closest to 20, which are 16 (4^2) and 25 (5^2). The square roots of these numbers are 4 and 5, respectively. Taking the average of these two square roots, we get (4 + 5) / 2 = 4.5. Therefore, the square root of 20 is approximately 4.5.
2. Long Division Method
The long division method is another technique you can use to find the square root of a non-perfect square. This method involves dividing the given number by a number that is close to its square root and then adjusting the quotient until you reach the desired level of accuracy.
To use the long division method, follow these steps:
1. Write the given number under the long division symbol, with the decimal point aligned with the decimal point in the number.
2. Find the largest integer whose square is less than or equal to the leftmost part of the number. Write this integer as the first digit of the quotient.
3. Multiply the first digit of the quotient by the divisor and write the result below the number.
4. Subtract the result from the leftmost part of the number and bring down the next digit.
5. Repeat steps 2-4 until you have found the desired level of accuracy.
For example, to find the square root of 20 using the long division method, you would perform the following steps:
“`
 4.472
 ______
 20.000
 -16.000
 ______
   4.000
   -3.632
 ______
   0.368
“`
Therefore, the square root of 20 is approximately 4.472.
3. Newton’s Method
Newton’s method, also known as the Newton-Raphson method, is an iterative technique that can be used to find the square root of a non-perfect square. This method is based on the idea of approximating the square root by repeatedly refining an initial guess.
To use Newton’s method, follow these steps:
1. Choose an initial guess for the square root. A good starting point is the nearest integer whose square is less than or equal to the given number.
2. Calculate the square of the guess and subtract it from the given number.
3. Divide the result by twice the guess.
4. Add the result to the guess to get a new approximation.
5. Repeat steps 2-4 until the desired level of accuracy is achieved.
For example, to find the square root of 20 using Newton’s method, you would perform the following steps:
1. Choose an initial guess, such as 4.
2. Calculate the square of the guess: 4^2 = 16.
3. Subtract the square from the given number: 20 – 16 = 4.
4. Divide the result by twice the guess: 4 / (2 4) = 0.5.
5. Add the result to the guess: 4 + 0.5 = 4.5.
Now, repeat steps 2-5 with the new guess (4.5):
1. Calculate the square of the guess: 4.5^2 = 20.25.
2. Subtract the square from the given number: 20 – 20.25 = -0.25.
3. Divide the result by twice the guess: -0.25 / (2 4.5) = -0.0222.
4. Add the result to the guess: 4.5 – 0.0222 = 4.4778.
Repeat the process until the desired level of accuracy is achieved. In this case, the square root of 20 is approximately 4.472136.
In conclusion, finding the square root of a non-perfect square can be done using various methods, such as estimation, long division, and Newton’s method. Each method has its own advantages and can be used depending on the level of accuracy required and the available resources.
