Question

If ${(z^3–3)}^2+16=0$ then the value of $\left|z\right|$ is

Answer 1

Step 1: Simplify the given expression

The given expression is,

${(z^3–3)}^2+16=0$

$\Rightarrow$ ${(z^3–3)}^2=-16$

$\Rightarrow$ ${(z^3–3)}^2=16i^2$ $\left[\because i^2=-1\right]$

$\Rightarrow$ ${(z^3–3)}^2={\left(4i\right)}^2$

$\Rightarrow$ $z^3–3=\pm 4i$

$\Rightarrow$ $z^3=3\pm 4i$

Step 2: Evaluate the modulus

Taking the modulus on both sides we get

${\left|z\right|}^3=\left|3\pm 4i\right|$

We know that $\left|z\right|=\sqrt{x^2+y^2}$

$\therefore {\left|z\right|}^3=\sqrt{3^2+4^2}$

$=\sqrt{9+16}$

$=\sqrt{25}$

$=5$

$\therefore \left|z\right|=5^{\frac{1}{3}}$

Hence, the required value of $\left|z\right|$ is $5^{\frac{1}{3}}$.