Question

Find the real and complex solution of the equation $x^4-256=0$.

Short answer

The real and solutions are $x=\pm 4$and complex solutions are $x=\pm 4i$.

Step-by-step solution

Step 1. Solve the equation.

$\begin{array}{c}{x}^4-256=\mathrm{0....}\left(\text{Given}\right)\\ \text{or,}\left(x^2-16\right)\left(x^2+16\right)=0\\ x^2-16=0\text{or}{x}^2+16=0\\ x^2=16\text{or}{x}^2=-\mathrm{16....}\left(1\right)\end{array}$

Step 2. By concept of complex numbers and factorizing.

Further Solving for (1),

$\begin{array}{c}x=\pm \sqrt{16}\text{or}x=\pm \sqrt{-16}\\ x=\pm 4\text{or}x=\pm 4i\end{array}$

Step 3. DESCRIPTION OF STEPS.

Since the roots are real and complex hence it has the solution $x=\pm 4,\pm 4i$.