Chapter I, Problem 20E

To determine

To find: The solution of the equation $x^4=1$.

Answer to Problem 20E

The solutions of the equation $x^4=1$ is $x=\pm 1$ and $x=\pm i$.

Explanation of Solution

Simplify the expression $x^4=1$ as follows.

$\begin{array}{l}{x}^4=1\\ x^4-1=0\\ {\left(x^2\right)}^2-1^2=0\\ \left(x^2-1\right)\left(x^2+1\right)=0\end{array}$

Further simplified as,

$\begin{array}{l}\left(x^2-1\right)=0\text{or}\left(x^2+1\right)=0\\ x^2=1\text{or}{x}^2=-1\\ x=\pm 1\text{or}x=\pm i\end{array}$

Thus, the solutions of the equation $x^2+2x+5=0$ is $x=\pm 1$ and $x=\pm i$.