Answer
Verified
285.6k+ views
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.
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.
Recently Updated Pages
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Find the values of other five trigonometric ratios class 10 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Find the angle in radian through which a pendulum swings class 10 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE