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

**Roots of a complex number**

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 has two different roots, the difference between the roots will be .

Hence the second root of will be

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 in polar form.”

**Solution**

Now here the complex number is . First of all, I’ll give it a name, say, .

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 . Let’s say, is the argument of .

Then the value of will be

So this means

Now if I compare with , I can say that the sign of is negative but the sign of is positive. So this means the value of will be in between and degrees.

(2)

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

Since the value of is , I can say the value of is

(3)

And this is the polar from of the complex number .

Next, I’ll get the fifth roots of .

**Step 2**

Since has five different roots, I can get the first root of from equation (3).

Thus it will be

Now I’ll give it a name, say .

So this means

As has five different roots, the difference between the roots will be

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 in polar form are , , , and .

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 , giving the results in the form of .”

##### Solution

Now here the number is . So, in the complex number form, it will be . First of all, I’ll give it a name, say, .

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 . Let’s say, is the argument of .

Then the value of will be

So this means

Now if I compare with , I can say that the sign of is negative but the sign of is positive. So this means the value of will be in between and degrees.

(5)

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

(6)

And this is the polar from of the complex number .

Next, I’ll get the fourth roots of .

**Step 2**

Since has four different roots, I can get the first root of from equation (6).

Thus it will be

Now I’ll give it a name, say .

So this means

Now I’ll simplify it to get

As has four different roots, the difference between the roots will be

Also, I know that .

So the second root will be

If I simplify it, then it becomes

Now I’ll substitute the values of and in to get the value of as

So this gives

Next, the third root will be

Then I’ll simplify it to get

Now I’ll substitute the values of and in to get the value of as

So this gives

Finally, the fourth root will be

Then I’ll simplify it to get

Now I’ll substitute the values of and in to get the value of as

Next, I’ll simplify it to get

Hence I can conclude that the fourth roots of in polar form are , , and .

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 , giving the results in polar form.”

##### Solution

Now here the number is . So, in the complex number form, it will be . First of all, I’ll give it a name, say, .

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 . Let’s say, is the argument of .

Then the value of will be

So this means

Now if I compare with , I can say that the sign of is negative but the sign of is positive. So this means the value of will be in between and degrees.

(8)

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

(9)

And this is the polar from of the complex number .

Next, I’ll get the fifth roots of .

**Step 2**

Since has five different roots, I can get the first root of from equation (9).

Thus it will be

Now I’ll give it a name, say .

So this means

As has five different roots, the difference between the roots will be

So the second root will be

Now I’ll simplify it to get the value of as

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 as

And this means

Hence I can conclude that the fifth roots of in polar form are , , , and .

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!!

Parth says

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