- Square Root of a Complex Number


Find \(\sqrt{15+8i}\)

We assume that the square root of a complex number is a complex number
  • Square both sides and simplify
  • Equate real parts
\(a^2-b^2=15\) (1)
  • Equate imaginary parts
\(\displaystyle a={4\over b}\) (2)
  • Substitute (2) into (1) and solve for \(a\) and \(b\)
\(\displaystyle \left({4\over b}\right)^2-b^2=15\)
\(\displaystyle {16\over b^2} -b^2=15\)
\(b^2=-16\) Invalid since \(b\) is real
  • Substitute values for a and b into a+bi
\(\sqrt{15+8i}=4+i\) or \(-4-i\)
Thus, we see that a complex number has 2 square roots, which are complex numbers.