Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 4 } + 16 = 0$$
A) $\left\{ 2 \operatorname { cis } 45 ^ { \circ } , 2 \operatorname { cis } 315 ^ { \circ } , 2 \right.$ cis $135 ^ { \circ } , 2$ cis $\left. 225 ^ { \circ } \right\}$
B) $\left\{ \sqrt { 3 } \operatorname { cis } 26.565 ^ { \circ } , \sqrt { 3 } \operatorname { cis } 333.435 ^ { \circ } , \sqrt { 3 } \operatorname { cis } 153.435 ^ { \circ } , \sqrt { 3 } \operatorname { cis } 206.565 ^ { \circ } \right\}$
C) $\left\{ 16 \operatorname { cis } 45 ^ { \circ } , 16 \operatorname { cis } 315 ^ { \circ } , 16 \right.$ cis $135 ^ { \circ } , 16$ cis $\left. 225 ^ { \circ } \right\}$
D) $\left\{ \sqrt { 2 } \operatorname { cis } 45 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 315 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 135 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 225 ^ { \circ } \right\}$

Question

Quizplus · Accepted Answer

The equation $x^4 + 16 = 0$ can be rewritten as $x^4 = -16$. Taking the fourth root of both sides gives $x = \sqrt[4]{-16}$. Since $-16$ can be written as $16(\cos 180^\circ + i\sin 180^\circ)$, the fourth roots are found by dividing the angle by 4 and finding all four quadrantal roots. This results in angles of $45^\circ$, $135^\circ$, $225^\circ$, and $315^\circ$, with a magnitude of $\sqrt[4]{16} = 2$. Therefore, the solutions are $2\operatorname{cis}45^\circ$, $2\operatorname{cis}135^\circ$, $2\operatorname{cis}225^\circ$, and $2\operatorname{cis}315^\circ$.

Find All Solutions of the Equation x4+16=0x ^ { 4 } + 16 = 0x4+16=0

Find All Solutions of the Equation $x ^ { 4 } + 16 = 0$