Find the formulae for the given expressions after simplifying it.
(i) \[{{a}^{3}}-{{b}^{3}}\]
(ii) \[{{a}^{3}}+{{b}^{3}}\]
(iii) \[{{\left( a+b \right)}^{3}}\]
(iv) \[{{\left( a-b \right)}^{3}}\]

Hint: To solve the expression \[\left( {{a}^{3}}-{{b}^{3}} \right)\], just add and subtract the terms \[{{a}^{2}}b\] and \[a{{b}^{2}}\] . Now, arrange the expression as \[{{a}^{2}}\left( a-b \right)+ab\left( a-b \right)+{{b}^{2}}\left( a-b \right)\] . Then, take the term \[\left( a-b \right)\] as common and factorize the expression. Similarly, to solve the expression \[\left( {{a}^{3}}+{{b}^{3}} \right)\], just add and subtract the terms \[{{a}^{2}}b\] and \[a{{b}^{2}}\] . Now, arrange the expression as \[{{a}^{2}}\left( a+b \right)-ab\left( a+b \right)+{{b}^{2}}\left( a+b \right)\] . Then, take the term \[\left( a+b \right)\] as common and factorize the expression. Now, to solve the expression \[{{\left( a+b \right)}^{3}}\] , write it as the product of the terms \[{{\left( a+b \right)}^{2}}\] and \[\left( a+b \right)\] . Use the formula \[{{\left( x+y \right)}^{2}}={{x}^{2}}+2xy+{{y}^{2}}\] and expand the term \[{{\left( a+b \right)}^{2}}\] . Then, multiply the terms \[\left( {{a}^{2}}+2ab+{{b}^{2}} \right)\] and \[\left( a+b \right)\] , and then solve it further. Now, to solve the expression \[{{\left( a-b \right)}^{3}}\] , write it as the product of the terms \[{{\left( a-b \right)}^{2}}\] and \[\left( a-b \right)\] . Use the formula \[{{\left( x-y \right)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}\] and expand the term \[{{\left( a-b \right)}^{2}}\] . Then, multiply the terms \[\left( {{a}^{2}}-2ab+{{b}^{2}} \right)\] and \[\left( a-b \right)\] , and then solve it further.

Complete step by step solution:
According to the question, we have four expressions. We have to simplify those four given expressions.
In part (i), we have to simplify the expression,
\[{{a}^{3}}-{{b}^{3}}\] ……………………….(1)
Adding and subtracting the term \[{{a}^{2}}b\] in equation (1), we get
\[{{a}^{3}}+{{a}^{2}}b-{{a}^{2}}b-{{b}^{3}}\] ……………………..(2)
Now, adding and subtracting the term \[a{{b}^{2}}\] in equation (2), we get
\[{{a}^{3}}+{{a}^{2}}b-{{a}^{2}}b+a{{b}^{2}}-a{{b}^{2}}-{{b}^{3}}\] ……………………(3)
Arranging the terms in equation (3), we get
\[={{a}^{3}}-{{a}^{2}}b+{{a}^{2}}b-a{{b}^{2}}+a{{b}^{2}}-{{b}^{3}}\]
\[={{a}^{2}}\left( a-b \right)+ab\left( a-b \right)+{{b}^{2}}\left( a-b \right)\] ………………………………(4)
Now, taking the term \[\left( a-b \right)\] in equation (4), we get
\[=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\] .
Therefore, the formula for \[\left( {{a}^{3}}-{{b}^{3}} \right)\] after simplifying it is \[\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\] .
So, \[\left( {{a}^{3}}-{{b}^{3}} \right)=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\] ……………………(5)
In part (ii), we have to simplify the expression,
\[{{a}^{3}}+{{b}^{3}}\] ……………………….(6)
Adding and subtracting the term \[{{a}^{2}}b\] in equation (6), we get
\[{{a}^{3}}+{{a}^{2}}b-{{a}^{2}}b+{{b}^{3}}\] ……………………..(7)
Now, adding and subtracting the term \[a{{b}^{2}}\] in equation (7), we get
\[{{a}^{3}}+{{a}^{2}}b-{{a}^{2}}b+a{{b}^{2}}-a{{b}^{2}}+{{b}^{3}}\] ……………………(8)
Arranging the terms in equation (8), we get
\[={{a}^{3}}+{{a}^{2}}b-{{a}^{2}}b-a{{b}^{2}}+a{{b}^{2}}+{{b}^{3}}\]
\[={{a}^{2}}\left( a+b \right)-ab\left( a+b \right)+{{b}^{2}}\left( a+b \right)\] ………………………………(9)
Now, taking the term \[\left( a+b \right)\] in equation (9), we get
\[=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\] .
Therefore, the formula for \[\left( {{a}^{3}}+{{b}^{3}} \right)\] after simplifying it is \[\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\] .
So, \[\left( {{a}^{3}}+{{b}^{3}} \right)=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\] ……………………(10)
In part (iii), we have to simplify the expression,
\[{{\left( a+b \right)}^{3}}\] ……………………………..(11)
We can write \[{{\left( a+b \right)}^{3}}\] as the product of the terms \[{{\left( a+b \right)}^{2}}\] and \[\left( a+b \right)\] .
Now, on transforming equation (11), we get
\[{{\left( a+b \right)}^{3}}={{\left( a+b \right)}^{2}}\times \left( a+b \right)\] ………………………..(12)
We know the formula, \[{{\left( x+y \right)}^{2}}={{x}^{2}}+2xy+{{y}^{2}}\] ………………………….(13)
Using equation (13) and transforming equation (12), we get
\[{{\left( a+b \right)}^{3}}=\left( {{a}^{2}}+2ab+{{b}^{2}} \right)\times \left( a+b \right)\] ………………………………(14)
Now, multiplying the terms \[\left( {{a}^{2}}+2ab+{{b}^{2}} \right)\] and \[\left( a+b \right)\] of equation (14), we get
\[\begin{align}
  & {{\left( a+b \right)}^{3}}=\left( {{a}^{2}}+2ab+{{b}^{2}} \right)\times \left( a+b \right) \\
 & {{\left( a+b \right)}^{3}}={{a}^{2}}.a+2ab.a+{{b}^{2}}.a+{{a}^{2}}.b+2ab.b+{{b}^{2}}.b \\
 & {{\left( a+b \right)}^{3}}={{a}^{3}}+2{{a}^{2}}b+a{{b}^{2}}+{{a}^{2}}b+2a{{b}^{2}}+{{b}^{3}} \\
\end{align}\]
\[{{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}\] …………………………………….(15)
Simplifying equation (15), we get
\[{{\left( a+b \right)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right)\] .
Therefore, the formula for \[{{\left( a+b \right)}^{3}}\] after simplifying it is \[\left\{ {{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right) \right\}\] .
So, \[{{\left( a+b \right)}^{3}}=\left\{ {{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right) \right\}\] ……………………………(16)
In part (iv), we have to simplify the expression,
\[{{\left( a-b \right)}^{3}}\] ……………………………..(17)
We can write \[{{\left( a-b \right)}^{3}}\] as the product of the terms \[{{\left( a-b \right)}^{2}}\] and \[\left( a-b \right)\] .
Now, on transforming equation (17), we get
\[{{\left( a-b \right)}^{3}}={{\left( a-b \right)}^{2}}\times \left( a-b \right)\] ………………………..(18)
We know the formula, \[{{\left( x-y \right)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}\] ………………………….(19)
Using equation (19) and transforming equation (18), we get
\[{{\left( a-b \right)}^{3}}=\left( {{a}^{2}}-2ab+{{b}^{2}} \right)\times \left( a-b \right)\] ………………………………(20)
Now, multiplying the terms \[\left( {{a}^{2}}-2ab+{{b}^{2}} \right)\] and \[\left( a-b \right)\] of equation (20), we get
\[\begin{align}
  & {{\left( a-b \right)}^{3}}=\left( {{a}^{2}}-2ab+{{b}^{2}} \right)\times \left( a-b \right) \\
 & {{\left( a-b \right)}^{3}}={{a}^{2}}.a-2ab.a+{{b}^{2}}.a-{{a}^{2}}.b+2ab.b-{{b}^{2}}.b \\
 & {{\left( a-b \right)}^{3}}={{a}^{3}}-2{{a}^{2}}b+a{{b}^{2}}-{{a}^{2}}b+2a{{b}^{2}}-{{b}^{3}} \\
\end{align}\]
\[{{\left( a-b \right)}^{3}}={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}}\] …………………………………….(21)
Simplifying equation (21), we get
\[{{\left( a-b \right)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right)\] .
Therefore, the formula for \[{{\left( a-b \right)}^{3}}\] after simplifying it is \[\left\{ {{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right) \right\}\] .
So, \[{{\left( a-b \right)}^{3}}=\left\{ {{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right) \right\}\] …………………………(23)
Hence, we have got the formulas from equation (5), equation (10), equation (16), and equation (22), we have
\[\left( {{a}^{3}}-{{b}^{3}} \right)=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\] ,
\[\left( {{a}^{3}}+{{b}^{3}} \right)=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)\] ,
\[{{\left( a+b \right)}^{3}}=\left\{ {{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right) \right\}\] ,
\[{{\left( a-b \right)}^{3}}=\left\{ {{a}^{3}}-{{b}^{3}}-3ab\left( a-b \right) \right\}\] .

Note: In this question, one might get confused while solving the expressions \[\left( {{a}^{3}}-{{b}^{3}} \right)\] and \[\left( {{a}^{3}}+{{b}^{3}} \right)\] because we don’t have any formulas so that we can use it here and solve these two expressions. The only way to solve these two expressions is to add and subtract the terms \[{{a}^{2}}b\] and \[a{{b}^{2}}\] in the expression \[\left( {{a}^{3}}+{{b}^{3}} \right)\] .