Question

Simplify the term ${\left( {{{\text{a}}^2} - {{\text{b}}^2}} \right)^2}$
$
  {\text{A}}{\text{. }}{\left( {\text{b}} \right)^2}{\left( {{\text{a - b}}} \right)^2} \\
  {\text{B}}{\text{. }}{\left( {{\text{a + b}}} \right)^2}{\left( {{\text{a - b}}} \right)^2} \\
  {\text{C}}{\text{. }}{\left( {{\text{a - b}}} \right)^2}{\left( {{\text{a - b}}} \right)^2} \\
  {\text{D}}{\text{. }}{\left( {{\text{a + b}}} \right)^2}{\left( {\text{a}} \right)^2} \\
$

Answer 1

Hint – To find the answer, we expand the given term and we expand each of the terms individually given in the choices to verify and find which term is equal to the given.

Complete step-by-step answer:

Given Data, ${\left( {{{\text{a}}^2} - {{\text{b}}^2}} \right)^2}$
It is of the form${\left( {{\text{x - y}}} \right)^2} = {{\text{x}}^2} + {{\text{y}}^2} - {\text{2xy}}$, where x = ${{\text{a}}^2}$and y =${{\text{b}}^2}$. Hence we get
$ \Rightarrow {\left( {{{\text{a}}^2}} \right)^2} - 2{{\text{a}}^2}{{\text{b}}^2} + {\left( {{{\text{b}}^2}} \right)^2}$
$ \Rightarrow {{\text{a}}^4} - {\text{2}}{{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4}$

Similarly, ${\left( {{\text{x + y}}} \right)^2} = {{\text{x}}^2} + {{\text{y}}^2}{\text{ + 2xy}}$
Now we expand the give options:
$
  {\text{A}}{\text{.}}{\left( {\text{b}} \right)^2}{\left( {{\text{a - b}}} \right)^2} \\
  {\text{ }} \Rightarrow {{\text{b}}^2}\left( {{{\text{a}}^2} + {{\text{b}}^2} - 2{\text{ab}}} \right) \\
  {\text{ }} \Rightarrow {{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4} - {\text{2a}}{{\text{b}}^3} \\
$
This is not equal to ${{\text{a}}^4} - {\text{2}}{{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4}$

$
  {\text{B}}{\text{.}}{\left( {{\text{a + b}}} \right)^2}{\left( {{\text{a - b}}} \right)^2} \\
  {\text{ }} \Rightarrow \left( {{{\text{a}}^2} + {{\text{b}}^2} + 2{\text{ab}}} \right)\left( {{{\text{a}}^2} + {{\text{b}}^2} - 2{\text{ab}}} \right) \\
  {\text{ }} \Rightarrow {{\text{a}}^4} + 2{{\text{a}}^2}{{\text{b}}^2}{\text{ - 2}}{{\text{a}}^3}{\text{b + }}{{\text{b}}^4} - {\text{2a}}{{\text{b}}^3} + {\text{2}}{{\text{a}}^3}{\text{b}} + {\text{2a}}{{\text{b}}^3} - 4{{\text{a}}^2}{{\text{b}}^2} \\
  {\text{ }} \Rightarrow {{\text{a}}^4} - {\text{2}}{{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4} \\
$
This is equal to ${{\text{a}}^4} - {\text{2}}{{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4}$. Hence Option B is an answer.
$
  {\text{C}}{\text{.}}{\left( {{\text{a - b}}} \right)^2}{\left( {{\text{a - b}}} \right)^2} \\
  {\text{ }} \Rightarrow \left( {{{\text{a}}^2} + {{\text{b}}^2} - 2{\text{ab}}} \right)\left( {{{\text{a}}^2} + {{\text{b}}^2} - 2{\text{ab}}} \right) \\
  {\text{ }} \Rightarrow {{\text{a}}^4} + 2{{\text{a}}^2}{{\text{b}}^2}{\text{ - 2}}{{\text{a}}^3}{\text{b + }}{{\text{b}}^4} - {\text{2a}}{{\text{b}}^3}{\text{ - 2}}{{\text{a}}^3}{\text{b - 2a}}{{\text{b}}^3} + 4{{\text{a}}^2}{{\text{b}}^2} \\
  {\text{ }} \Rightarrow {{\text{a}}^4}{\text{ + 6}}{{\text{a}}^2}{{\text{b}}^2} - 4{\text{a}}{{\text{b}}^3} - 4{{\text{a}}^3}{\text{b}} + {{\text{b}}^4} \\
$
This is not equal to ${{\text{a}}^4} - {\text{2}}{{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4}$

$
  {\text{D}}{\text{.}}{\left( {\text{a}} \right)^2}{\left( {{\text{a + b}}} \right)^2} \\
  {\text{ }} \Rightarrow {{\text{a}}^2}\left( {{{\text{a}}^2} + {{\text{b}}^2} + 2{\text{ab}}} \right) \\
  {\text{ }} \Rightarrow {{\text{a}}^2}{{\text{b}}^2} + {{\text{a}}^4}{\text{ + 2}}{{\text{a}}^3}{\text{b}} \\
$
This is not equal to ${{\text{a}}^4} - {\text{2}}{{\text{a}}^2}{{\text{b}}^2} + {{\text{b}}^4}$

Hence Option B is the correct answer.

Note – In order to solve this type of problems the key is to know the formulas ${\left( {{\text{x + y}}} \right)^2} = {{\text{x}}^2} + {{\text{y}}^2}{\text{ + 2xy}}$ and ${\left( {{\text{x - y}}} \right)^2} = {{\text{x}}^2} + {{\text{y}}^2} - {\text{2xy}}$ which are used in the expansion.
This problem can be solved in a simpler way, i.e. by substituting random values in place of a and b for the given equation and all the choices. Whichever gives the value equal to the given equation is the solution.