Solve the equation by using substitution. $z ^ { 4 } - 256 = 0$
A) $\{ 4 , - 4 \}$
B) $\{ 4 , - 4,16 i , - 16 i \}$
C) $\{ 4 \}$
D) $\{ 4 , - 4,4 i , - 4 i \}$

Question

Quizplus · Accepted Answer

This equation is a quartic equation that can be solved by substitution. First, notice that $z^4 - 256 = 0$ can be rewritten as $z^4 = 256$. This can be seen as $(z^2)^2 = 256$, suggesting a substitution of $y = z^2$, leading to $y^2 = 256$. Solving for $y$, we get $y = 16$ or $y = -16$. Substituting back for $z^2$, we have $z^2 = 16$ and $z^2 = -16$. Solving $z^2 = 16$ gives $z = 4$ or $z = -4$, and solving $z^2 = -16$ gives $z = 4i$ or $z = -4i$. Therefore, the solution set includes $4, -4, 4i$, and $-4i$.

Solve the Equation by Using Substitution z4−256=0z ^ { 4 } - 256 = 0z4−256=0

Solve the Equation by Using Substitution $z ^ { 4 } - 256 = 0$