Solve the following expression, ${{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}}$.

Hint: We are given an expression of the form ${{a}^{2}}-{{b}^{2}}$. So, we can solve this by using ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ by taking a = 205 and b = 195 and then solve it to get the value of ${{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}}$.

Complete step-by-step solution
In this question, we have been asked to find the value of the expression, ${{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}}$. Let us consider the expression given in the question as follows.
$\Rightarrow E={{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}}.........\left( i \right)$
Let 205 = a and 195 = b. By substituting these value in equation (i), we will get,
$\Rightarrow E={{a}^{2}}-{{b}^{2}}$
By adding and subtracting the term (ab) on the RHS of the above equation, we will get,
$\Rightarrow E={{a}^{2}}-{{b}^{2}}+ab-ab$
By grouping the terms of the above equation, we will get,
$\Rightarrow E=\left( {{a}^{2}}+ab \right)-\left( {{b}^{2}}+ab \right)$
Now, by taking out the common terms from each group we will get,
$\Rightarrow E=a\left( a+b \right)-b\left( a+b \right)$
By taking out the term (a + b) common from both the terms, we get,
$\Rightarrow E=\left( a+b \right)\left( a-b \right)$
Now, by again substituting a = 205 and b = 195 that we had assumed earlier, in the above equation, we will get,
$\begin{align}
  &\Rightarrow E=\left( 205+195 \right)\left( 205-195 \right) \\
 &\Rightarrow E=\left( 400 \right)\times \left( 10 \right) \\
 &\Rightarrow E=4000 \\
\end{align}$
Therefore, we get the value of the expression, that is, ${{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}}$ as 4000.

Note: In this question, the students must remember that any expression of the form ${{a}^{2}}-{{b}^{2}}$ is solved by writing it as $\left( a+b \right)\left( a-b \right)$. In this question, some students may find ${{\left( 205 \right)}^{2}}$ and ${{\left( 195 \right)}^{2}}$ separately and then solve the question as follows:
$\begin{align}
  &\Rightarrow E={{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}} \\
 &\Rightarrow E=42025-38025 \\
 &\Rightarrow E=4000 \\
\end{align}$
From this also, we get ${{\left( 205 \right)}^{2}}-{{\left( 195 \right)}^{2}}=4000$. Though this method is also correct, it is very time consuming because of the multiplication of 3 digit numbers. And also it can lead to calculation mistakes which results in the wrong answer. So, the students must choose the method accordingly and then solve the question correctly.