Question

Find the square root of $-7-24\sqrt{-1}$.

Answer 1

Assume
$\sqrt{-7-24\sqrt{-1}}=x+y\sqrt{-1}$
then $-7-24\sqrt{-1}=x^2-y^2+2xy\sqrt{-1}$
$\therefore x^2-y^2=-\mathrm{7....}\left(1\right)$
and $2xy=-24$
$\therefore {(x^2+y^2)}^2={(x^2-y^2)}^2+{\left(2xy\right)}^2$
$=49+576$
$=625$
$\therefore x^2+y^2=\mathrm{25.....}\left(2\right)$
From (1) and (2), $x^2=9;y^2=16$
$\therefore x=\pm 3,y=\pm 4$
Since the product $xy$ is negative, we must take
$x=3,y=-4$; or $x=-3,y=4$
Thus the roots are $3-4\sqrt{-1}$ and $-3+4\sqrt{-1}$;
that is $\sqrt{-7-24\sqrt{-1}}=\pm (3-4\sqrt{-1})$