hi all,
i need help with two problems that read: "solve the equation in the complex number system."
&
$\displaystyle x^4 = 256$
$\displaystyle (x^2)^2 = 256$
$\displaystyle x^2 = \pm 16$.
Case 1: $\displaystyle x^2 = 16$
$\displaystyle x = \pm 4$.
Case 2: $\displaystyle x^2 = -16$
$\displaystyle x = \pm 4i$.
So the four solutions are $\displaystyle 4, 4i, -4, -4i$.
The second expression, $\displaystyle x^3- 343$, isn't and equation but I assume you meant $\displaystyle x^3= 343$. Since the first equation had a simple integer solution, x= 2, the first thing you should do is start cubing integers. You find that $\displaystyle 7^3= 343$. Then $\displaystyle x^3- 7^3= (x- 7)(x^2+ 7x+ 49)$. x= 7 is one solution and you can find the other two by solving $\displaystyle x^2+7x+ 49= 0$ with the quadratic formula.