Question

Find all the complex roots of the equation x4 − 16 = 0.

Answer 1

4+16=0

4=−16=162

2=±4

Case 1 2=+4

=±2√

√=±(12√+12√)

=±2‾√+2‾√

Case 2 2=−4=43

=±23/2=±2()1/2

∵√=±(12√+12√)

=±(2‾√+2‾√)

So in all,

=2‾√+2‾√

=−2‾√+2‾√

=2‾√−2‾√

=−2‾√−2‾√

Let √=+

Squaring both sides, we have

=(+)2

0+1=2−2+2⟹

2−2=0(1)

2=1

=12

Plugging in (1) … ⟹

2−1(2)2=0

4=14

2=12

∴=±12‾√

∴=±12‾√

∴√=+=±(12+12)

This can also be shown as below

√=12+12

Or

√=−12−12 \