FIND THE SQUARE ROOT OF COMPLEX NUMBER

Find the square root of complex number :

Here we are going to see how to find the square root of complex number.

Let a + ib be a complex number such that √a + ib  =  x + iy.

Let us look in to some example problems to understand the concept.

Example 1 :

Find the square root of the following 

7 - 24i

Solution :

Let √7 - 24i  =  x + iy, then

√(7 - 24i)  =  x + iy

Taking squares on both sides

7 - 24i  =  (x + iy)²

7 - 24i  =  x² + (iy)² + 2(x)(iy)

7 - 24i  =  x² - y² + i 2xy

x² - y²  =  7   ------(1)     2xy  =  -24  ------(2)

We have a formula,

(x² + y²)²  =  (x² - y²)² + 4x²y²

=  (x² - y²)² + (2xy)²

=  7² + (-24)²

 (x² + y²)²  =  49 + 576  ==>  625

 (x² + y²)  =  √625  ==>  25

x² + y²  =  25    ------(3)

(1)  +  (3) ==>   x² - y² + x² + y²  =  25 + 7

2x²  =  32

x²  =  16 ==>  x  =  ± 4

Hence the square root of the given complex number are

4 - 3i or (-4 + 3i)

Example 2 :

Find the square root of the following 

-15 - 8i

Solution :

Let √-15 - 8i  =  x + iy, then

√(-15 - 8i)  =  x + iy

Taking squares on both sides

-15 - 8i  =  (x + iy)²

-15 - 8i  =  x² + (iy)² + 2(x)(iy)

-15 - 8i  =  x² - y² + i 2xy

x² - y²  =  -15  ------(1)     2xy  =  -8  ------(2)

We have a formula,

(x² + y²)²  =  (x² - y²)² + 4x²y²

=  (x² - y²)² + (2xy)²

=  (-15)² + (-8)²

 (x² + y²)²  =  225 + 64  ==>  289

 (x² + y²)  =  √289  ==>  17

x² + y²  =  17    ------(3)

(1)  +  (3) ==>   x² - y² + x² + y²  =  -15 + 17 

2x²  =  2

x²  =  1 ==>  x  =  ± 1

Hence the square root of the given complex number are

1 - 4i or (-1 + 4i)

Example 3 :

Find the square root of the following 

-8 - 6i

Solution :

Let √-8 - 6i  =  x + iy, then

√(-8 - 6i)  =  x + iy

Taking squares on both sides

-8 - 6i  =  (x + iy)²

-8 - 6i  =  x² + (iy)² + 2(x)(iy)

-8 - 6i  =  x² - y² + i 2xy

x² - y²  =  -8  ------(1)     2xy  =  -6  ------(2)

We have a formula,

(x² + y²)²  =  (x² - y²)² + 4x²y²

=  (x² - y²)² + (2xy)²

=  (-8)² + (-6)²

 (x² + y²)²  =  64 + 36  ==>  100

 (x² + y²)  =  √100  ==>  10

x² + y²  =  10    ------(3)

(1)  +  (3) ==>   x² - y² + x² + y²  =  -8 + 10 

2x²  =  2

x²  =  1 ==>  x  =  ± 1

Hence the square root of the given complex number are

1 - 3i or (-1 + 3i)

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.