Use the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$, find the value of ${209^2}$. Also, evaluate the square of \[97\]using the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.

Hint: In the given question, we have to evaluate a square of a number given to us in the problem itself with the help of algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ and ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ . The algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ is used to evaluate the square of a binomial expression involving the sum of two terms. Similarly, the identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ is used to evaluate the square of a binomial expression involving the difference of two terms. In order to solve the problem, we split up the numbers whose squares are to be computed in such a way that the calculations become easier and then apply the algebraic identities to get to the final answer.

Complete step by step answer:
Given question requires us to find the value of a square of $209$. We are asked to use the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ for the task described. So, to calculate the square of $209$, we have to first divide the number $209$ into two parts such that the following calculation of the square of the number becomes easier. So, we know that $209 = 200 + 9$. So, we can divide $209$ into the two numbers $200$ and $9$. So, we now substitute the values of the two parts into the algebraic identity that we are supposed to use. Hence, we have, ${209^2} = {\left( {200 + 9} \right)^2}$

Now, we expand the left side of the equation using the algebraic identity to evaluate the square of a binomial expression involving the sum of two terms. So, we get,
$ \Rightarrow {209^2} = {\left( {200} \right)^2} + 2\left( {200} \right)\left( 9 \right) + {\left( 9 \right)^2}$
$ \Rightarrow {209^2} = 40000 + 3600 + 81$
$ \Rightarrow {209^2} = 43681$
So, the value of ${209^2}$ calculated using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$is $43681$.
Also, we have to evaluate the square of \[97\]using the algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.
So, to calculate the square of \[97\], we have to first divide the number \[97\] into two parts such that the following calculation of the square of the number becomes easier.

So, we know that $97 = 100 - 3$. So, we can express \[97\] as a difference of two numbers: $100$ and $3$. So, we now substitute the values of the two parts into the algebraic identity that we are supposed to use.Hence, we have, ${97^2} = {\left( {100 - 3} \right)^2}$. Now, we expand the right side of the equation using the algebraic identity to evaluate the square of a binomial expression involving the difference of two terms. So, we get,
${97^2} = {\left( {100} \right)^2} - 2\left( {100} \right)\left( 3 \right) + {\left( 3 \right)^2}$
Simplifying the expression further, we get,
$ \Rightarrow {97^2} = 10000 - 600 + 9$
$ \therefore {97^2} = 9409$

Note:Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. We can also verify the answer of the given question by calculating the squares of 209 and 97 simply. The squares of the numbers can be calculated in many ways by splitting up the numbers into two different parts each time.