**How to find Square root of a complex number**

A number of the form where and is called a complex number where is called as real part and is called imaginary part of complex number. To find the square root of a complex number, we will assume that . Then square both the sides and compare real and imaginary part to find the value of and , which will give us the square root.

**Formula for finding square root of a complex number:**

The square root of is

for and

for

**Proof. **Let square root of . That is where and in . Now square both the sides we get

Equating real and imaginary parts we get

Now

Solving (i) and (iii) we get

.

Similarly . Since 2ab=y , it is clear that both and have the same sign when is positive and and have different sign when is negative. Therefore

for and

for

**Example : Find the square root of (-5-12i) .**

**Sol. **Here

Applying the formula for square root we get

( is negative)

**Trick to find the square root of a complex number: **

To find , follow the following steps:

- First find the number .
- Now factorise the given number in such a way that difference of square of these factors is equal to the real number .

**Ex. Find the square root of 7+4i.**

Sol. Find the number which is Equal to 12. Now factor 12 in such a way that difference of square of these factors is equal to the real number .

12= 4×3 and . Therefore .

**Ex. Find the square root of .**

Sol. Here and such that

Thus

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