Answer
The simplest form of the expression ${{\left( {{x}^{2}}-3y \right)}^{4}}$ is ${{x}^{8}}+81{{y}^{4}}+54{{x}^{4}}{{y}^{2}}-12{{x}^{6}}y-108{{x}^{2}}{{y}^{3}}$.
Work Step by Step
Consider the expression
${{\left( {{x}^{2}}-3y \right)}^{4}}$,
The above expression can be written as,
${{\left( {{x}^{2}}-3y \right)}^{2}}{{\left( {{x}^{2}}-3y \right)}^{2}}$
Apply the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$,
$\begin{align}
& {{\left( {{x}^{2}}-3y \right)}^{2}}={{\left( {{x}^{2}} \right)}^{2}}+{{\left( 3y \right)}^{2}}-2\left( {{x}^{2}} \right)\left( 3y \right) \\
& ={{x}^{4}}+9{{y}^{2}}-6{{x}^{2}}y
\end{align}$
Substitute the value of ${{\left( {{x}^{2}}-3y \right)}^{2}}$ in ${{\left( {{x}^{2}}-3y \right)}^{2}}{{\left( {{x}^{2}}-3y \right)}^{2}}$
$\begin{align}
& {{\left( {{x}^{2}}-3y \right)}^{2}}{{\left( {{x}^{2}}-3y \right)}^{2}}=\left( {{x}^{4}}+9{{y}^{2}}-6{{x}^{2}}y \right)\left( {{x}^{4}}+9{{y}^{2}}-6{{x}^{2}}y \right) \\
& ={{x}^{4}}\left( {{x}^{4}}+9{{y}^{2}}-6{{x}^{2}}y \right)+9{{y}^{2}}\left( {{x}^{4}}+9{{y}^{2}}-6{{x}^{2}}y \right)-6{{x}^{2}}y\left( {{x}^{4}}+9{{y}^{2}}-6{{x}^{2}}y \right) \\
& ={{x}^{8}}+9{{x}^{4}}{{y}^{2}}-6{{x}^{6}}y+9{{x}^{4}}{{y}^{2}}+81{{y}^{4}}-54{{x}^{2}}{{y}^{3}}-6{{x}^{6}}y-54{{x}^{2}}{{y}^{3}}+36{{x}^{4}}{{y}^{2}}
\end{align}$
Group together the like terms and solve them,
$\begin{align}
& {{\left( {{x}^{2}}-3y \right)}^{4}}={{x}^{8}}+9{{x}^{4}}{{y}^{2}}-6{{x}^{6}}y+9{{x}^{4}}{{y}^{2}}+81{{y}^{4}}-54{{x}^{2}}{{y}^{3}}-6{{x}^{6}}y-54{{x}^{2}}{{y}^{3}}+36{{x}^{4}}{{y}^{2}} \\
& ={{x}^{8}}+81{{y}^{4}}+9{{x}^{4}}{{y}^{2}}+9{{x}^{4}}{{y}^{2}}+36{{x}^{4}}{{y}^{2}}-6{{x}^{6}}y-6{{x}^{6}}y-54{{x}^{2}}{{y}^{3}}-54{{x}^{2}}{{y}^{3}} \\
& ={{x}^{8}}+81{{y}^{4}}+54{{x}^{4}}{{y}^{2}}-12{{x}^{6}}y-108{{x}^{2}}{{y}^{3}}
\end{align}$
Therefore, the simplest form of the expression ${{\left( {{x}^{2}}-3y \right)}^{4}}$ is ${{x}^{8}}+81{{y}^{4}}+54{{x}^{4}}{{y}^{2}}-12{{x}^{6}}y-108{{x}^{2}}{{y}^{3}}$.
