Evaluate \[{\left( {53} \right)^2}\] by using the identity \[{\left( {a{\text{ }} + {\text{ }}b} \right)^2} = {\text{ }}{a^2} + {\text{ }}2ab{\text{ }} + {\text{ }}{b^2}.\]

Hint: We can simplify \[53\] as \[\left( {50 + 3} \right).\] Now, we will compare with the given identity, we will take a \[ = {\text{ }}50\] and b \[ = {\text{ }}3.\] And solve the equation further. This way by using identity we will get the value of \[{\left( {53} \right)^2}.\]

Complete step-by-step solution:
The given identity is \[{\left( {a{\text{ }} + {\text{ }}b} \right)^2} = {\text{ }}{a^2} + {\text{ }}2ab{\text{ }} + {\text{ }}{b^2}.\]
\[53\] can be written as \[\left( {50 + 3} \right).\]
\[\begin{gathered}
  \left( {50{\text{ }} + {\text{ }}3} \right){\text{ }} = {\text{ }}{\left( {50} \right)^2} + {\text{ }}2503{\text{ }} + {\text{ }}{\left( 3 \right)^2} \\
  \begin{array}{*{20}{l}}
  { = {\text{ }}2500{\text{ }} + {\text{ }}300{\text{ }} + {\text{ }}9} \\
  { = {\text{ }}2809}
\end{array} \\
\end{gathered} \]
Therefore, \[{\left( {53} \right)^2} = {\text{ }}2809\]

Note: By using identity, it is easier to evaluate the value of \[{\left( {53} \right)^2}.\] While solving these types of question, students should always keep in mind to simplify the number in such a way, where it is easier to double the number.