Evaluate: ${\left( {101} \right)^2}$.

Hint: We can find the square of a number using many methods, but we are going to use the mathematical identity to find its square. The identity we will be using is
$ \to {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$
Here, we need to express 101 as a sum of two numbers and then use the above identity.

Complete step-by-step answer:
In this question, we have to find the square of 101 using the identity.
Now, we can find the square using many different methods like log method, but in this question, we are going to use the mathematical identity to find the square of a number.
The identity is
$ \to {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$
So, we are going to use this identity to find the square of 101.
For that, we need to express 101 as a sum of two numbers. Now, we could do this in many ways like $\left( {97 + 4} \right)$, $\left( {98 + 3} \right)$ but we need to express it in such a way that we do not need to use the calculator. So, we will be expressing 101 as $\left( {100 + 1} \right)$.
Hence, $a = 100$ and $b = 1$. So, using the identity we will get
$ \to {\left( {101} \right)^2} = {\left( {100 + 1} \right)^2}$
                 $
   = {\left( {100} \right)^2} + 2\left( {100} \right)\left( 1 \right) + {\left( 1 \right)^2} \\
   = 10000 + 200 + 1 \\
   = 10000 + 201 \\
   = 10201 \\
 $
Hence, the square of 101 is 10201.

Note: We can also find the square of 101 using another property, that is ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.
Here, we need to express 101 as a difference of two numbers instead of sum of two numbers. So, we can express 101 as $\left( {110 - 9} \right)$. Hence, using the identity we will get,
$ \to {\left( {101} \right)^2} = {\left( {110 - 9} \right)^2}$
                 $
   = {\left( {110} \right)^2} - 2\left( {110} \right)\left( 9 \right) + {\left( 9 \right)^2} \\
   = 12100 - 1980 + 81 \\
   = 12100 - 1899 \\
   = 10201 \\
 $
Hence, the square of 101 is 10201.