Question

How do you find the square root of a complex number?

Answer 1

Hint: A complex number is a number that can be expressed in the form of sum of a real number and imaginary number. A square root of a number is nothing but the number that is being multiplied twice. So let us find out the formula for finding the square root of a complex number.

Complete step-by-step solution:
Now let us briefly discuss complex numbers. We know that complex numbers are the combination of real numbers and imaginary numbers. Imaginary numbers, upon squaring, give a negative number. In a complex number, either the real part or the imaginary part can be zero. We can perform addition and multiplication upon the complex numbers. The conjugate of a complex number results in the change of the mathematical sign. Normally, a conjugate of a number is written with a bar over the number. A complex number is represented in the form of \[a+ib\] where a.b are real.
Now let us find the square root of a complex number.
The formula for expressing the square root of a complex number is:
\[\sqrt{a+ib}=\pm \sqrt{\dfrac{\left| z \right|+a}{2}}+i\dfrac{b}{\left| b \right|}\sqrt{\dfrac{\left| z \right|-a}{2}}\]

Note: While finding the square root of a complex number, we must have a note that \[a+ib\] and \[b\] are not equal to zero. In some of the cases, we can also find the square root of a complex number by expressing it in the form of algebraic identities.