The complex equation $x^3 = 9 + 46i$ has a solution of the form $a + bi$ where $a,b\in \mathbb Z$. Find the value of $a^3 + b^3$ The complex equation $x^3 = 9 + 46i$ has a solution of the form $a + bi$ where $a,b\in \mathbb Z$. Find the value of $a^3 + b^3$ .

Let $x=a+bi$, where $\{a,b\}\subset\mathbb R$.
Thus, $$(a+bi)^3=9+46i$$ or
$$a^3-3ab^2=9$$ and
$$3a^2b-b^3=46,$$
which gives
$$46(a^3-3ab^2)=9(3a^2b-b^3)$$ or
$$(2a+3b)(23a^2-48ab+3b^2)=0,$$ which gives
$$2a+3b=0.$$
Thus, $$3\cdot\frac{9}{4}b^3-b^3=46,$$ which gives $b=2$, $a=-3$ and $a^3+b^3=-19.$
