Square Root by Long Division Method

The square root of a number is a value that, when squared, results in the given number. For example, the square root of 25 is ±5, as 5 × 5 = 25. One of the methods used to find the square root is the long division method. The long division method involves components such as the dividend, divisor, quotient, and remainder. This method involves the process of dividing, multiplying, subtracting, bringing down, and repeating. 
 

There are different methods to find the square root of a number, such as the long division and factorization methods. In this section, we will learn the difference between the long division method and the factorization method. 

Now let’s learn how to find the square root of a number using the long division method. Here we will find the square root of 2025 using the long division method.

Step 1: First, we group the numbers (dividend) from right to left in pairs. Here, the number is 2025; it can be grouped as 20 and 25.

Step 2: Find a number (first digit of the quotient) whose square is less than or equal to 20. 4² = 16, which is less than 20. Now, subtract 16 from 20 to get the remainder as 4.

Step 3: Now we bring down the second pair, so the new dividend is 425

Step 4: Double the current quotient (4) to get 8, then append a digit x to form the new divisor as 8x.

Step 5: Find the value of x such that \(8x × x ≤ 425\). Here, x = 5, so the divisor is \(85 (85 × 5 = 425)\). So, 5 is the next digit in the quotient and 0 is the remainder. 

As there is no remainder, the quotient is 45. Therefore, the square root of, 2025 is ±45. 
 

When the square root of a number is a whole number, then the number is a perfect square. We use the long division method to find the square root of a perfect square. Let's learn with an example, finding the square root of 1521.

• First, we pair the number 1521. Here the first pair is 15, and the second pair is 21

• Finding a number whose square is less than or equal to 15. As 32 = 9, the first digit of the quotient is 3. Subtracting 15 and 9, 15 – 9 = 6.

• Bringing down the second pair so the new division is 621

• Double the current quotient (3) to get 6. Then append a digit x to form the new divisor as 6x.

• Finding the value of x in 6x, the value of \(6x × x \) should be less than or equal to 621

• Here, the value of x is 9, because \(69 × 9 = 621\). So, the new quotient is 9

• The final quotient is 39, so the square root of 1521 is ±39. So, the square of 39 and -39 equals 1521.

The number that does not have a whole number as its square root is the non-perfect square root. Let’s find the square root of a non-perfect square number with an example.


Finding the square root of 11
 

Identify the perfect square closer to the given number, and split the number into the nearest perfect square. \(√11 = √(9 + 2)\)


Find the difference between the given number and the nearest perfect square, here 11 – 9 = 2, then divide that difference by twice the square root of the perfect square and add or subtract the result
\(√11 = √9 + 2\), 


Using approximation formula: \(\sqrt{a + b} \approx \sqrt{a} + \frac{b}{2\sqrt{a}} \)
\(\sqrt{11} = 3 + \frac{2}{2 \times 3} \)
=\(3 + \frac{2}{6} \)
= 3 + 0.333 
= 3.333

Using the approximation formula: \(\sqrt{a - b} \approx \frac{\sqrt{a} - b}{2\sqrt{a}} \) 

\(\sqrt{9 + 3} \)
 

The long division method is used to find the square root of any number. This method has some advantages and disadvantages; in this section, we will discuss some advantages and disadvantages of the long division method. 
 

The long division method helps find the square root of a number step by step by grouping digits and estimating carefully. Regular practice improves speed and accuracy.

• Always group the digits of the number in pairs starting from the decimal point, moving both left and right.
   
• Find the largest number whose square is less than or equal to the first group and subtract its square.
   
• Bring down the next pair of digits and double the current quotient to find the next digit of the root.
   
• Estimate carefully when selecting the next digit to avoid errors and ensure accuracy.
   
• Practice with different numbers to become confident and faster with the long division square root method.

To find the square root of a number we use the long division method used in different fields like geometry, physics, math, etc. Let’s understand some real-world applications of the square root by division method. 

• To calculate the distance between two points in geometry, we use the distance formula and it involves square root. To solve the square root we use the long division method. 

• To find the length of the square when we have the area, we use the long division method. As the area of the square is a2. 

• In physics many formulas involve square roots such as kinetic energy, velocity value, speed in free fall, etc.(v = 2Km, T = 2πLg, v = 2gh), we use a long division method.