Question

The expression $(x^2+y^2{)}^4$ is equal to

• A. $(x^4-2x^2{y}^2+y^4{)}^2+(4x^{3}y+4x{y}^3{)}^2$
• B. $(x^4+6x^2{y}^2-y^4{)}^2+(4x^{3}y-4x{y}^3{)}^2$
• C. $(x^4-6x^2{y}^2+y^4{)}^2+(4x^{3}y-4x{y}^3{)}^2$
• D. None of these

Answer 1

The correct option is C $(x^4-6x^2{y}^2+y^4{)}^2+(4x^{3}y-4x{y}^3{)}^2$

$(x^2+y^2{)}^4=|x+iy{|}^8$
$={\left|\right(x+iy{)}^2|}^4$
$={\left|\right(x^2-y^2)+2ixy|}^4$
$={\left|{\left[\right(x^2-y^2)+2ixy]}^2\right|}^2$
$={\left|\right(x^2-y^2{)}^2+(2ixy{)}^2+2(x^2-y^2\left)\right(2ixy\left)\right|}^2$
$={|x^4+y^4-2x^2{y}^2-4x^2{y}^2+i(4x^{3}y-4x{y}^3\left)\right|}^2$
$={|x^4+y^4-6x^2{y}^2+i(4x^{3}y-4x{y}^3)|}^2$
$={(x^4-6x^2{y}^2+y^4)}^2+{(4x^{3}y-4x{y}^3)}^2$