Question

Find the square roots of the following numbers by Prime factorization method.
(i) 441
(ii) 784
(iii) 4096
(iv) 7056

Answer 1

Hint: To find the square root of a given number by prime factorization method, we will start dividing the number by the smallest prime number which is a factor of the given number until we get all the factors of the number.

Complete step-by-step solution -
Prime factorization is done by writing all the factors of a number which are prime. For, ex 6 can be written as 6 = 2 x 3, where 2 and 3 are prime numbers. So, we have to first write all the prime factors of given numbers and then find their square root.
First, we will find the factors of 441. So, we will start dividing 441 by the smallest prime number. Here, 3 is the smallest prime number which is also a factor of 441. So, we get
${\text{441 = 3}} \times {\text{3}} \times 7 \times 7$
Taking square root both sides, we get
$\sqrt {441} {\text{ = }}\sqrt {{\text{3}} \times {\text{3}} \times 7 \times 7} $
A square of any number is formed by multiplying that number to itself. So, to get a square root, we will take a pair of factors. So, we get
$\sqrt {441} {\text{ = }}\sqrt {{\text{(3}} \times {\text{3)}} \times (7 \times 7)} $ = 3 x 7 = 21
Therefore, $\sqrt {441} {\text{ = 21}}$
Now, we will find the square root of 784. We will prime factorize 784 and will write all the factors of it.
Therefore, 784 = 2 x 2 x 2 x 2 x 7 x 7
Now, taking square root both sides, we get
$\sqrt {784} {\text{ = }}\sqrt {2 \times 2 \times 2 \times 2 \times 7 \times 7} $. Pairing the factors to get the square root, we get
$\sqrt {784} {\text{ = }}\sqrt {(2 \times 2) \times (2 \times 2) \times (7 \times 7)} $ = 2 x 2 x 7 = 4 x 7 = 28
Therefore, $\sqrt {784} {\text{ = 28}}$
Now, we will find the square root of 4096. We will prime factorize 4096 and will write all the factors of it.
So, 4096 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Now, taking square root both sides, we get
\[\sqrt {4096} {\text{ = }}\sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2} \]. Pairing the factors to get the square root, we get
\[\sqrt {4096} {\text{ = }}\sqrt {(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)} \] = 2 x 2 x 2 x 2 x 2 x 2 = 64
Therefore, $\sqrt {4096} {\text{ = 64}}$
Now, we will find the square root of 7056. We will prime factorize 7056 and will write all the factors of it.
7056 = 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7
Now, taking square root both sides, we get
$\sqrt {7056} {\text{ = }}\sqrt {2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7} $. Pairing the factors to get the square root, we get
$\sqrt {7056} {\text{ = }}\sqrt {(2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)} $ = 2 x 2 x 3 x 7 = 84
Therefore, \[\sqrt {7056} {\text{ = 84}}\]
Therefore, $\sqrt {441} {\text{ = 21}}$
$\sqrt {784} {\text{ = 28}}$
$\sqrt {4096} {\text{ = 64}}$
\[\sqrt {7056} {\text{ = 84}}\]

Note: When we come up with such types of questions, we have to find the factors of the given number in the prime number. Prime factorization method is used to break the given number into a product of prime numbers. To find the square root, we have to take the pair of factors, while to find the cube root, we have to take the triplet of factors.