Question

Simplify ${{\left( {{m}^{2}}-{{n}^{2}}m \right)}^{2}}+2{{m}^{2}}{{n}^{2}}$

Answer 1

Hint: use algebraic identity of ${{\left( a-b \right)}^{2}}$with the given expression. Identity is given as ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$

Complete step-by-step answer:

Use property of surds while simplifying the expression after applying the above mentioned identity. Some properties are given as
$\begin{align}
  & {{m}^{a}}.{{m}^{b}}={{m}^{a+b}} \\
 & {{\left( {{m}^{a}} \right)}^{b}}={{m}^{ab}} \\
 & {{\left( mn \right)}^{ab}}={{m}^{a}}{{n}^{b}} \\
\end{align}$

Given expression in the problem is
$={{\left( {{m}^{2}}-{{n}^{2}}m \right)}^{2}}+2{{m}^{3}}{{n}^{2}}$
Now, as we know the algebraic identity of ${{\left( a-b \right)}^{2}}$, that can be given as
 ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$…………… (ii)
Hence, we can use this identity with the given expression in equation (i), with the term ${{\left( {{m}^{2}}-{{n}^{2}}m \right)}^{2}}$where we can suppose value of ‘a’ as ${{m}^{2}}$ and ‘b’ as ${{n}^{2}}m$ for the equation (ii). Hence, we can get equation (i) as
 $={{\left( {{m}^{2}}-{{n}^{2}}m \right)}^{2}}+2{{m}^{3}}{{n}^{2}}$
$={{\left( {{m}^{2}} \right)}^{2}}+{{\left( {{n}^{2}}m \right)}^{2}}-2\times {{m}^{2}}\times {{n}^{2}}m+2{{m}^{3}}{{n}^{2}}$…………… (iii).
As, we know the property of surds, that are given as
$\begin{align}
  & {{\left( {{A}^{c}} \right)}^{d}}={{A}^{cd}}........................\left( iv \right) \\
 & {{\left( AB \right)}^{c}}={{A}^{c}}{{B}^{c}}...................\left( v \right) \\
 & {{A}^{c}}{{A}^{d}}={{A}^{c+d}}....................\left( vi \right) \\
\end{align}$
Hence, we can get equation (iii) with the help of above properties of surds 2x – 3 =3(x – 3) with the help of second condition in the problem. One may write the equation as 3(2x -3) = x – 3, which is wrong. So, be careful and don’t confuse with the words. Hence, the terms of the expression (iii) can be simplified as ${{\left( {{m}^{2}} \right)}^{2}}$ can be simplified with the help of equation (iv) as
${{\left( {{m}^{2}} \right)}^{2}}={{m}^{4}}$
${{\left( {{n}^{2}}m \right)}^{2}}$ can be simplified using equation (v) as
${{\left( {{n}^{2}}m \right)}^{2}}={{\left( {{n}^{2}} \right)}^{2}}{{m}^{2}}$
Now, we can further use equation (iv) so, we can re-write the above expression as
${{\left( {{n}^{2}}m \right)}^{2}}={{n}^{4}}{{m}^{2}}$
Third term of the expression (iii) can be given as
$2{{m}^{2}}\times {{n}^{2}}m=2{{m}^{2}}\times {{m}^{{}}}\times {{n}^{2}}$
Using equation (vi) we get
$2{{m}^{2}}\times {{n}^{2}}m=2{{m}^{3}}{{n}^{2}}$
Hence, equation (iii), can be re-written with the help of above equations as
$={{m}^{4}}+{{n}^{4}}{{m}^{2}}-2{{m}^{3}}{{n}^{2}}+2{{m}^{3}}{{n}^{2}}$
Now, we can observe that the terms $2{{m}^{3}}{{n}^{2}},-2{{m}^{3}}{{n}^{2}}$ can be cancelled out each other as both are equal and with opposite signs.
So, we get the value of simplified form of the given expression in the problem.
$={{m}^{4}}+{{n}^{4}}{{m}^{2}}$
Now, we can observe that the term ${{m}^{4}}$can also be written as ${{m}^{2}}\times {{m}^{2}}$ and hence, we can take ${{m}^{2}}$as common from both the terms of the above expression. So, we get
$={{m}^{2}}\left( {{m}^{2}}+{{n}^{4}} \right)$
Hence the simplified form of the given expression in the problem is $={{m}^{2}}\left( {{m}^{2}}+{{n}^{4}} \right)$.
So, $={{m}^{2}}\left( {{m}^{2}}+{{n}^{4}} \right)$is the required answer of the given question.
Don’t need to think about $\left| A \right|$ . Don’t confuse with the identity of
$\begin{align}
  & x={{A}^{-1}}B \\
 & x=\left[ \begin{matrix}
   x \\
   y \\
   z \\
\end{matrix} \right] \\
\end{align}$
One may use identity as $x=B{{A}^{-1}}$ which is wrong. So, be clear with the formula for future reference.

Note: One may multiply ${{m}^{2}}-{{n}^{2}}m$ twice to get the square of ${{m}^{2}}-{{n}^{2}}m$ without using the identity of ${{\left( a-b \right)}^{2}}$. Apply the property of surds carefully. So, take care of the property, while simplifying the given expression.