a). Evaluate \[\left( 493 \right)^{2}\] using identities.
b). Find the smallest number by which \[10368\] must be divided so that it becomes a square number , also find the square root of the resulting number.

Answer
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Hint: In this question, first we need to evaluate \[\left( 493 \right)^{2}\] using identities. Then we need to find the smallest number by which \[10368\] must be divided so that it becomes a square number , and also find the square root of the resulting number. First, we can find the factors of the number \[10368\]. Then we need to make the number \[10368\] as a perfect square number. In order to make the number a perfect square number, we need to divide the number by a factor with no pair . Then on simplifying, we will get the resulting number. Then on taking the square root of the resulting number, we will get our required answer.

Complete step-by-step answer:
First let us solve the part (a) .
Given , \[\left( 493 \right)^{2}\]
We can rewrite \[493\] as \[490 + 3\]
By using algebraic identity \[\left( a + b \right)^{2} = a^{2} + b^{2} + 2ab\]
Here \[a\] is \[490\] and \[b\] is \[3\]
On applying the identity,
We get,
\[\left( 490 + 3 \right)^{2} = 490^{2} + 3^{2} + 2\left( 490 \right)\left( 3 \right)\]
On simplifying,
We get,
\[\Rightarrow \ 240100 + 9 + 2940\]
On further simplifying,
We get,
\[\Rightarrow \ 243,049\]
Thus we get the value of \[\left( 493 \right)^{2}\] is \[243,049\] .
Now let us solve part (b)
Given, \[10368\]
First let us find the factors of the number \[10368\] ,
\[10368 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\]
The number \[2\] in the factors of \[10368\] is without a pair. So, if we divide \[10368\] by \[2\] , we get a perfect square.
\[\Rightarrow \dfrac{10368}{2}\]
On simplifying,
We get,
\[\Rightarrow \ 5184\]
Thus we get the number \[5184\] which is a perfect square number.
Now let us find the square root of \[5184\] .
\[\Rightarrow \ \sqrt{5184} = \sqrt{8 \times 8 \times 9 \times 9}\]
On taking terms out of radical sign,
We get,
\[\Rightarrow \ 8 \times 9\]
On simplifying,
We get,
\[\Rightarrow \ 72\] .
Thus the square root of \[5184\] is \[72\] .
Hence we get the smallest number is \[2\] by which \[10368\] are divided so that the result is a perfect square \[5184\] and the square root of the resulting number is \[72\] .

Final answer :
a). The value of \[\left( 493 \right)^{2}\] is \[243,049\] .
b). The smallest number is \[2\] by which \[10368\] are divided so that the result is a perfect square \[5184\] and the square root of the resulting number is \[72\].

Note: The concept used to solve these types of questions is the prime factorisation method . We can also use a calculator to simplify the final step. If we are using the calculator to find out the answer , we can simply enter the square root of \[5184\] which will give us the correct answer. We should be very careful while taking the numbers out of the radical sign.