Roots of a complex number. Today it’s all about the roots of a complex number. Have a look!!

**Roots of a complex number**

If you are looking for more in complex numbers, do check in:

**Addition and subtraction of complex numbers**

**Multiplication and division of complex numbers**

**Polar form of a complex number**

**Modulus and argument of a complex number**

**Functions of complex variables**

Suppose is a complex number. And I have to find out the second roots of this complex number.

Now for that, first of all, I’ll write it in the polar form.

Say, is its polar form.

So the first root of will be

If

Hence the second root of

And it will continue in this way.

Now I’ll give some examples of how to find out the roots of a complex number.

**Solved examples of the roots of a complex number**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example of the roots of a complex number.

**Example 1**

According to Stroud and Booth (2013)*, “Find the fifth roots of

**Solution**

Now here the complex number is

So I can say

Now I’ll find out all the fifth roots of

So I’ll start with the polar form of

**Step 1**

First of all, I’ll find out the modulus of

And that is

Next, I’ll simplify it to get

And this means

(1)

Now I’ll get the argument of

Then the value of

So this means

Now if I compare

(2)

Next, I’ll combine equations (1) and (2) to get the polar form of

Since the value of

(3)

And this is the polar from of the complex number

Next, I’ll get the fifth roots of

**Step 2**

Since

Thus it will be

Now I’ll give it a name, say

So this means

As

So the second root will be

If I simplify it, then it becomes

Then the third root will be

After simplification it becomes

Now the fourth root will be

Next, I’ll simplify it to get

Finally, the fifth root will be

Then I’ll simplify it to get

Hence I can conclude that the fifth roots of

And this is the answer to this example.

Now I’ll give another example of the roots of a complex number.

##### Example 2

According to Stroud and Booth (2013)*, “Determine the fourth roots of

##### Solution

Now here the number is

So I can say

Now I’ll find out all the fourth roots of

So I’ll start with the polar form of

**Step 1**

First of all, I’ll find out the modulus of

And that is

(4)

Now I’ll get the argument of

Then the value of

So this means

Now if I compare

(5)

Next, I’ll combine equations (4) and (5) to get the polar form of

(6)

And this is the polar from of the complex number

Next, I’ll get the fourth roots of

**Step 2**

Since

Thus it will be

Now I’ll give it a name, say

So this means

Now I’ll simplify it to get

As

Also, I know that

So the second root will be

If I simplify it, then it becomes

Now I’ll substitute the values of

So this gives

Next, the third root will be

Then I’ll simplify it to get

Now I’ll substitute the values of

So this gives

Finally, the fourth root will be

Then I’ll simplify it to get

Now I’ll substitute the values of

Next, I’ll simplify it to get

Hence I can conclude that the fourth roots of

And this is the answer to this example.

Now I’ll give another example of the roots of a complex number.

##### Example 3

According to Stroud and Booth (2013)*, “Find the fifth roots of

##### Solution

Now here the number is

So I can say

Now I’ll find out all the fifth roots of

So I’ll start with the polar form of

**Step 1**

First of all, I’ll find out the modulus of

And that is

(7)

Now I’ll get the argument of

Then the value of

So this means

Now if I compare

(8)

Next, I’ll combine equations (7) and (8) to get the polar form of

(9)

And this is the polar from of the complex number

Next, I’ll get the fifth roots of

**Step 2**

Since

Thus it will be

Now I’ll give it a name, say

So this means

As

So the second root will be

Now I’ll simplify it to get the value of

Next, the third root will be

So I’ll simplify it to get

Then the fourth root will be

So this gives

Finally, I’ll get the fifth root of

And this means

Hence I can conclude that the fifth roots of

And this is the answer to this example.

Dear friends, this is the end of today’s post on the roots of a complex number. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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