# 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.

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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.

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 $72$ .

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.