Question

Find the square root of the following:
(a) 1444
(b) 1849
(c) 5776
(d) 7921

Answer 1

Hint: You need to use either of the two methods for finding the square root of a given number, namely prime factorisation in which we break the number as the product of its prime factors and try to make pairs of its factors to reach the answer, while the other is long division method which comes to play when it is difficult to break a number as the product of its prime factors. So, use either of the two methods, whichever you find suitable to reach the answer.

Complete step-by-step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. Once we have represented the number as the product of primes, just make pairs of prime factors and take square root to get the answer. We will be discussing the division method while using it in the question itself, which would enhance your understanding.

(a) 1444
For finding the square root of 1444, we will use the method of prime factorisation. As 1444 is an even number, we can represent this as: $1444=2\times 722$ . 722 is again an even number, so we will write it as: $1444=2\times 2\times 361$ . Also, we know that 361 is the square of 19, as it is important to remember the squares till 30.
$1444=2\times 2\times 19\times 19$
Now we will take square root of both the sides:
$\sqrt{1444}=\sqrt{2\times 2\times 19\times 19}$
$\Rightarrow \sqrt{1444}=2\times 19=38$
Therefore, the square root of 1444 is 38.

(b) 1849
For numbers like 1849, we will use the method of prime factorization. So, first, we will express 1849 as the product of prime numbers.
Now, 1849 is an odd number. So, 1849 is not divisible by 2. Also, the sum of the digits is not a multiple of 3, so 1849 is not divisible by 3 and 9 as well. So, if we keep on checking each prime number in the increasing order starting from 2 for finding the factor of 1849, we will find that 43 is a factor of 1849. So, 1849 can be written as $1849=43\times 43$ . So, the square root of 1849 can be written as $\sqrt{1849}=43$ . Hence, the square root of 1849 is 43.

(c) 5776
Now, coming to the next question, we will use the method of prime factorization. So, first we will express 5776 as the product of prime numbers.
Now, 5776 is an even number. So, we can write 5776 as $5776=2\times 2888$ . 2888 is also an even number. So, we can write 2888 as $2888=2\times 1444$ . We know, 1444 is the root of 38 from part (a). So, we can write 1444 as $1444=38\times 38$ . So, $5776=2\times 2\times 38\times 38$ . So, the square root of 5776 can be written as $\sqrt{5776}=2\times 38=76$ . Hence, the square root of 5776 is 76.

(d) 7921
For numbers like 7921 where it is difficult to find the factors, we use the division method for finding the root. First, we make a pair of digits starting from the unit place. Then we need to find the largest perfect square, smaller than the number formed by the pair of digits appearing first from the right.
$8\overset{8}{\overline{\left){\begin{align}
  & 7921 \\
 & \underline{64\text{ }} \\
 & 1521 \\
\end{align}}\right.}}$
Now double the quotient you got in the first step and make it the tenth place of a number, now find a unit place to the number such that the number into its unit place is equal to or closest to the number you obtained as the remainder of the previous steps. Also, make the unit digit of the number, the unit digit of the quotient as well. Repeat the steps till you get the remainder zero.
\[\begin{align}
  & 8\overset{89}{\overline{\left){\begin{align}
  & 7921 \\
 & \underline{64\text{ }} \\
 & 169\overline{\left){1521}\right.} \\
 & \text{ }\underline{1521} \\
\end{align}}\right.}} \\
 & \text{ 0} \\
\end{align}\]
Therefore, the root of 7921 is 89.

Note: While calculating square roots and cube roots, prime factorization is the easiest method. But it is not possible to use it every time. Hence, other methods should also be learnt, so that they can be used while solving problems. Generally, for finding the numbers which are very large or decimals which have no prime factors, the method of long division is used.