Question

Let Z be a complex number satisfying the equation $(Z^3+3{)}^2=-16$ then |Z| has the value equal to

• A. $5^{\frac{1}{2}}$
• B. $5^{\frac{1}{3}}$
• C. $5^{\frac{2}{3}}$
• D. $5$

Answer 1

The correct option is B $5^{\frac{1}{3}}$

$(z^3+3{)}^2=-16$

=$(z^3+3{)}^2=16i^2$

$=(z^3+3)=4ior-4i$

$=z^3=4i-3or-4i-3$

Taking modulus both sides :

$\mid z{\mid }^3=\mid 4i-3\mid or\mid -4i-3\mid$

$\therefore \mid z{\mid }^3=5$

$\mid z\mid =5^{\frac{1}{3}}$