Question

find the square root of the \[9604\] by prime factorization method.

Answer 1

Hint:
We have given a number and have to find its square root by prime factorization method. For this we have to write prime factors of the number in the product'. After that, we have to do the pairing of the prime [number. We write one number from each pair to add the product to it. It will give us the required square root if all the prime numbers are paired then the number is a perfect square if not then the number I will not be a prime number square.

Complete step by step solution:
We have the number. 9604 we have to do the prime factorization of \[9604\].
The prime factor of \[9604\] are \[9604\]
\[9604{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}4802\]
\[4802{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}2401\]
\[2401{\text{ }} \div {\text{ }}7{\text{ }} = {\text{ }}343\]
\[343{\text{ }} \div {\text{ }}7{\text{ }} = {\text{ }}49\]
\[49{\text{ }} \div {\text{ }}7{\text{ }} = {\text{ }}7\]
\[7{\text{ }} \div {\text{ }}7{\text{ }} = {\text{ }}1\]

Now we have to find the square root.
 Taking square root on both sides
\[\left){\vphantom{1{9604}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{9604}}} = \left){\vphantom{1{2 \times 2 \times 7 \times 7 \times 7 \times 7}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{2 \times 2 \times 7 \times 7 \times 7 \times 7}}}\]
Now from the right-hand side, we have pairs of the prime number
\[2,7\] ,we will take one number from the pair
 Therefore \[\;\left){\vphantom{1{9604}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{9604}}} = 2 \times 7 \times 7 = 98\]

Therefore the square root \[19604\] is \[98\].

Note:
Prime number: The number which is divisible by itself and one only the number is called a prime number.
Some examples are 2, 3, 5, 7, 11, 13 etc.
Prime factorization: In number theory, Integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, The process is called Prime factorization.
Square root: In mathematics, a square root of a number x is a number y such as \[{y^2} = x\]