Question

Simplify the following : \[8{{a}^{3}}-{{b}^{3}}-12{{a}^{2}}b+6a{{b}^{2}}\]

Answer 1

Hint:Relate the given equation in the problem with the algebraic identity of \[{{\left( x-y \right)}^{3}}\]. It is given as \[{{\left( x-y \right)}^{3}}={{x}^{3}}-{{y}^{3}}-3{{x}^{2}}y+3x{{y}^{2}}\]
Compare the right hand side of the above equation and compare it with the given expression the problem. Try to get the solution.

Complete step-by-step answer:
Given expression in the problem is \[8{{a}^{3}}-{{b}^{3}}-12{{a}^{2}}b+6a{{b}^{2}}\] \[\to \] (1)
As we know the algebraic identity of \[{{\left( x-y \right)}^{3}}\], can be given as
\[{{\left( x-y \right)}^{3}}={{x}^{3}}-{{y}^{3}}-3{{x}^{2}}y+3x{{y}^{2}}\] \[\to \] (2)
Now, compare the right hand side of the above identity with the given expression in the equation (1).
So, we can write the given expression in the equation (1) as
\[8{{a}^{3}}-{{b}^{3}}-12{{a}^{2}}b+6a{{b}^{2}}\]
\[8{{a}^{3}}\] can be written as \[{{\left( 2a \right)}^{3}}\]
\[{{b}^{3}}\] can be written as \[{{\left( b \right)}^{3}}\]
\[12{{a}^{2}}b\] can be written as \[3{{\left( 2a \right)}^{2}}b\]
\[6a{{b}^{2}}\] can be written as \[3\left( 2a \right){{\left( b \right)}^{2}}\]
Hence, we get expression as
\[{{\left( 2a \right)}^{3}}-{{\left( b \right)}^{3}}-3{{\left( 2a \right)}^{2}}b+3\times 2a{{\left( b \right)}^{2}}\]
So, on comparing the expression in right hand side of the equation (2) and the above expression, we get
\[\begin{align}
  & x=2a \\
 & y=b \\
\end{align}\]
So, with the help of equation (1), we can write the expression given in the problem as
\[\begin{align}
  & {{\left( 2a-b \right)}^{3}}={{\left( 2a \right)}^{3}}-{{\left( b \right)}^{3}}-3{{\left( 2a \right)}^{2}}b+3\times 2a{{\left( b \right)}^{2}} \\
 & {{\left( 2a-b \right)}^{3}}=8{{a}^{3}}-{{b}^{3}}-12{{a}^{2}}b+6a{{b}^{2}} \\
\end{align}\]
Hence, the simplified form of the given expression is \[{{\left( 2a-b \right)}^{3}}\].

Note: One may use another formula of \[{{\left( x-y \right)}^{3}}\] as well that is given as \[{{\left( x-y \right)}^{3}}={{x}^{3}}-{{y}^{3}}-3xy\left( x-y \right)\]. As, it becomes the same identity when \[3xy\] gets multiplied with \[x\] and \[y\]. So, don’t confuse with the identities.Both of them are the same.Observation is the key point and relating the expression with some identity is also a key of the problem.