Find the value of ${{\left( 25 \right)}^{3}}$ using the short-cut method.

Hint: For these kinds of questions, we just need to apply some basic formulae. If required, we also need to perform some manipulations so as to get the answers. We can see a whole cube. We can do a manipulation here so as to use the whole cube formula. We can write $25$ as $\left( 20+5 \right)$ or $\left( 26-1 \right)$ . But writing it as $\left( 20+5 \right)$ is simpler while applying the ${{\left( a+b \right)}^{3}}$ . Since there is $20$ , we can easily find it’s cube rather than finding the cube of $26$.

Complete step by step solution:
So in this question, we will use ${{\left( a+b \right)}^{3}}$ formula and not ${{\left( a-b \right)}^{3}}$ .
Let us see the formula of ${{\left( a+b \right)}^{3}}$. It is the following :
$\Rightarrow {{\left( a+b \right)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right)$ .
We are going to write $25$ as $\left( 20+5 \right)$.
Let us write it.
\[~\Rightarrow {{\left( 25 \right)}^{3}}={{\left( 20+5 \right)}^{3}}\] .
Now let us compare what we have with the general form of the formula.
Upon doing so, our $a$ is $25$ and our $b$ is $5$.
Let us substitute and apply the formula.
Upon doing so, we get the following :
\[\begin{align}
  & ~\Rightarrow {{\left( 25 \right)}^{3}}={{\left( 20+5 \right)}^{3}} \\
 & ~\Rightarrow {{\left( 25 \right)}^{3}}={{\left( 20 \right)}^{3}}+{{\left( 5 \right)}^{3}}+3\times 20\times 5\left( 20+5 \right) \\
 & \Rightarrow {{\left( 25 \right)}^{3}}=8000+125+300\left( 25 \right) \\
 & \Rightarrow {{\left( 25 \right)}^{3}}=8125+7500 \\
 & \Rightarrow {{\left( 25 \right)}^{3}}=15625 \\
\end{align}\]
$\therefore $ The value of \[{{\left( 25 \right)}^{3}}\] upon using the short-cute method is $15625$.

Note: We can also write $25$ as $\left( 30-5 \right)$ since finding the cube of $30$ is also easier than finding the cube of $26$. After writing it that way, we should apply the ${{\left( a-b \right)}^{3}}$and we will get the same answer. So, to be able to solve problems of this kind, we should remember all these formulae. These formulae will not directly be asked as questions but so as to be able to simplify a trigonometric or algebraic equation, we need to make use of these formulae. We should be careful while calculating and substituting.