Square Root Long Division Method

Square Root is defined as the product of a number that is multiplied by itself. For example, the square root of 16 is 4, because 4 multiplied by 4 which is 16. Square Root Long Division Method is the method of calculating the square root of large numbers quickly. It is very useful for the students to understand how to calculate the Square Root with the Long Division Method because we can find the square root of any large number without using the prime factorization method or repeated subtraction method.

In this article, we have discussed, the Square Root Long Division Method with Examples, the Factorization Method of finding Square Roots, the Advantages and Disadvantages of Long Division Methods, and others in detail.

What is Long Division Method?

Long Division Method is a method of calculating the square root of a number which can be the product of prime numbers. It is the easiest and fastest way to calculate the square root of multiple-digit numbers. This includes dividends, divisors, quotients, and remainders.

The large digit number is the dividend divided by another number known as the divisor. The outcome of division is the quotient and the remaining part is the remainder. The example of finding a square root using the long division method is added in the image below,

How to Find Square Root by Using Long Division?

Square Root of any number can be easily found using the long division method by following the steps added below,

Step 1: Considering 4096 as a large number to find the square root. Mark the bar below the number by making a pair from the right side of the number. If we have the 3-digit number then the same process will be repeated, mark the bar on the single digit left also. For example 6 36

Step 2: Select a number (Divisor) whose square is less than or equal to the large number (Dividend). Divide it and write the quotient which is 6 and the remainder is 4.

Step 3: In this step, write down the remaining pair to the right side of the pair which is 96

Step 4: Now add the divisor with the quotient which is 6 + 6 = 12, and write down by leaving some space on the right side.

Step 5: Next, we select a new digit (which is 4) and write next to the new divisor which is 12 to find the number whose multiplication is less than or equal to the dividend which is 496.

Therefore, we get the quotient that is 64 which is the answer. So the square root of 4096 is 64.

Example of Square Root by Long Division Method

So in the below example, we have selected a large number 729 which is dividend. Do the pairing from the right side and then select a divisor which is whose square is less than or equal to the first pair? On solving and bringing down the remaining pair we get 329.

Now add the divisor with the quotient which is 2+2 = 4 then, we select a new digit (which is 7) and write next to the new divisor which is 4 to find the number whose multiplication is less than or equal to the new dividend which is 329.

Square Root of Perfect Square Number

A perfect square is a number that has a whole number in its Square Root. To find the square root of a perfect square by the long division method, we need to pair up the digits from the right and find the largest divisor and quotient that satisfy the division.

Let’s understand it with three simple steps

1. Make a pair of numbers from the right-hand side. Make sure that the first pair will have 2 numbers and the second pair will have the remaining numbers.
2. Now determine the square root that comes just before the first pair.
3. Multiply the square root with the next number and you will be able to confirm the unit of your square root.

Note: Perfect Square should never get end with 2,3,7,8 because we did not get the numbers ending with these digits in the below square.

Example: Solve √(2304)

Solution:

Step 1: Find in the square root table where do we get last digit 4. You’ll only get on 2 and 4.

Step 2: Now Select the first pair which is 23 and check perfect square of this number also less than of this number 23 which is 4 (4² = 16)

Step 3: Multiply the square root with the next number and you will be able to confirm the unit of your square root. Therefore, 4⨯5 = 20

√23 04 = ?

Possibility = 2, 8

Now,

4 ⨯ 5 = 20

Then, Possibility = 42 , 48 (Putting 4 before 2,8)

So, 23 is greater than 20 then our answer should be greater value which 48.

Square Root of Numbers 1 to 20

Square roots of numbers 1 to 20 are added in the table below,

12 = 1	112 = 121
22 = 4	122 = 144
32 = 9	132 = 169
42 = 16	142 = 196
52 = 25	152 = 225
62 = 36	162 = 256
72 = 49	172 = 289
82 = 64	182 = 324
92 = 81	192 = 361
102 = 100	202 = 400

Learn more about, Square Root 1 to 30

Square Root of 44100

Now let’s calculate the square root of 44100 using the long division method. The image added below shows the calculation for the same.

Square Root of Non-Perfect Square Number

A non-perfect square is a number that does not have a whole number as its Square Root. To find the square root of a non-perfect square by the long division method, we need to follow the same steps as for a perfect square.

Let’s understand it with just two simple steps,

Step 1: Split the number into the Perfect square closer to the given number, i.e.

√12 = √(9 + 3)

Step 2: Now we will write the perfect square value, whatever we have more or less divided by double of perfect square value

Now,

√(9 + 3) = 3 + 3/6 (3 is the perfect square value + perfect square value / double of perfect square)

= 3 + 0.5 = 3.5

Therefore, 3.5 is the answer

We can Solve this with other methods as,

√12

= √(16 – 4)

= 4 – 4/8

= 4 – 0.5 = 3.5 {4 (is the perfect square value) – (perfect square value/double of a perfect square)}

Long Division Vs Factorisation

Long division and prime factorization are two different methods of finding the factors of a number. Long division is a process of dividing a number by another number, usually a smaller one, to get a quotient and a remainder. Prime factorization is a process of finding the prime numbers that multiply together to get the original number. They have only two factors, 1 and the number itself.

Both methods give us the same result, but they use different approaches. Long division is more general and can be used to find any factor of a number, not just prime factors. Prime factorization is more specific and can be used to find the unique combination of prime numbers that make up a number.

Advantages of Long Division Method

Various advantages of the long division methods are,

• It helps to break down a complex division problem into simpler steps, making it easier to perform by hand.
• Long Division Method helps to understand the concepts of dividend, divisor, quotient, and remainder.
• It helps in finding the factors of a number, especially when the divisor is a prime number or a product of prime numbers.
• It helps to solve polynomial rational functions easily by dividing a polynomial by another polynomial of the same or lower degree.
• It helps to prepare for more complex topics in polynomial algebra, such as synthetic division, polynomial factorization, and finding roots.

Disadvantages of Long Division Method

Various disadvantages of the long division methods are,

• It can become complex when the number or the polynomial involved is large or has many terms. It requires a lot of steps and calculations, which can increase the chances of making errors.
• It can be challenging for some students, especially when there are missing terms, remainders, decimals, or fractions in the dividend or the divisor. So always have a good command of the basic operations.
• Long Division Method can be replaced by other methods that are simpler, faster, or more efficient in some cases, such as synthetic division, prime factorization, etc. It is not the best method for a particular situation or purpose

Read More,

• Square Root 1 to 100
• Square 1 to 30
• Sqauare and Square Roots

Problems on Square Roots by Long Division

Problem 1: Find the Square Root of 17424 by Long Division Method

Solution:

Square root of 17424 is found using the long division method as,

Thus, the square root of 17424 is 132.

Problem 2: Find the Square Root of 7.29 by Long Division Method

Solution:

In the decimal numbers, after getting the remainder which is 329 just mark the point after the quotient which is 2 that’s it. Now you can solve as it is with the same method.

Thus, the square root of 7.29 is 2.7.

Problem 3: Find the Square Root of 69169 by Long Division Method

Solution:

Square root of 69169 is found using the long division method as,

Thus, square root of 69169 is 263.

Problem 4: Find the Square Root of 27225 by Long Division Method

Solution:

Square root of 27225 is found using long division method as,

Thus, the square root of 27225 is 165.

Problem 5: Find the Square Root of 14161 by Long Division Method

Solution:

Square root of 14161 is found using the long division method as,

Thus, the square root of 14161 is 119.

Practice Problems on Square Roots by Long Division Method

P1. Evaluate the Square Root of 1444 by Long Division Method

P2. Find the Square Root of 54756 by Long Division Method

P3. Evaluate the Square Root of 3249 by Long Division Method

P4. Evaluate the Square Root of 2304 by Long Division Method

Square Roots by Long Division Method – FAQs

1. What are Square Roots?

Square roots are numbers, when multiplied by themselves, give the same number. For example, the square root of 16 is 4, because 4 times 4 are 16.

2. How to Find Square Root of a Number?

There are four methods to find the square root of a number, that are,

• Long Division Method
• Prime Factorization Method
• Repeated Subtraction Method
• Estimation Method

3. What is Long Division in Maths?

Long division in Maths is a way of finding the square root of a number by dividing it into smaller parts and using the quotient and remainder to get the answer by following the simple steps which are division, multiplication, subtraction, bring down and repeat.

4. How to Find Square Root of a Perfect Square by Long Division?

A perfect square is a number that has a whole number as its square root. To find the square root of a perfect square by long division method, we need to pair up the digits from the right and find the largest divisor and quotient that satisfy the division.

5. How to Find Square Root of a Non-Perfect Square by Long Division?

A non-perfect square is a number that does not have a whole number as its square root. To find the square root of a non-perfect square by long division method, we need to follow the same steps as for a perfect square but we have to add pairs of zeros or decimals to get the answer.

6. What is Prime Factorization Method?

Prime factorization is a method to find the prime factors of a number, such that the original number is evenly divisible by these factors. Prime numbers that can only be divided by 1 and themselves without leaving a leftover are known as prime factors.

7. How to Find Square Root of a Negative Number?

Square Root of Negative Number isn’t a real number, but a complex number. For example, the square root of -16 is 4i.

8. How to do Long Division?

Long division method is used to find the square root of the numbers. We follow the following steps to find the square root using long division.

Step 1: Check first digit of dividend from the left and check if this digit is greater than or equal to the divisor.

Step 2: Divide it by the divisor and find the quotient.

Step 3: Subtract the result as in normal division.

Step 4: Bring down the next digit of the dividend.

Step 5: Repeat from step 2 till we find the square root of required number.