Complex Numbers: Find the modulus of (1 + i sqrt{3}). Hence evaluate (1 + i sqrt{3})^3. Give your answer in x + iy form.

[reggiwilliams]

Been revising this topic and came across this question which I’m struggling with
Worked out answer to first part is 2 but unsure about 2nd and 3rd questions
Can someone steer me in right direction

 

[Dr.Peterson]

Been revising this topic and came across this question which I’m struggling with
Worked out answer to first part is 2 but unsure about 2nd and 3rd questions
Can someone steer me in right direction

I don't see a third question, unless perhaps they expect you (as the second question) to first evaluate the cube in polar form ... which may be a good idea, and uses the modulus. supporting the word "hence". (It's easy enough to evaluate it directly.)

 

[reggiwilliams]

View attachment 37412
I don't see a third question, unless perhaps they expect you (as the second question) to first evaluate the cube in polar form ... which may be a good idea, and uses the modulus. supporting the word "hence". (It's easy enough to evaluate it directly.)

Still struggling with this
Trust you agree the answer to first part is 2.
So would say this evaluates to 2 cubed ie 8
But do not know how to answer in form of x+yi?
Can you show how you arrive at this

 

[Dr.Peterson]

Still struggling with this
Trust you agree the answer to first part is 2.
So would say this evaluates to 2 cubed ie 8
But do not know how to answer in form of x+yi?
Can you show how you arrive at this

Yes, you are right about the modulus they ask for. But no, they aren't just asking for the modulus of the cube! (If the answer were 8, that would already be in x+yi form, unless you insisted on 8+0i.)

In order to be sure what they expect you to do, I'll need to know something about the context. What topic has been taught most recently (assuming this is from a course)? Do you know about polar form, and how to use it to raise a complex number to a power? (You may know it by a different name.)

And can you evaluate the cube directly, in order to see what the answer will be, and perhaps get an idea for what they want you to do?

The more you tell us about what you know, and what has been said about the problem, the quicker we can see how to help. At the very least, what course are you taking, and is this your first exposure to complex numbers?

 

[reggiwilliams]

Yes, you are right about the modulus they ask for. But no, they aren't just asking for the modulus of the cube! (If the answer were 8, that would already be in x+yi form, unless you insisted on 8+0i.)

In order to be sure what they expect you to do, I'll need to know something about the context. What topic has been taught most recently (assuming this is from a course)? Do you know about polar form, and how to use it to raise a complex number to a power? (You may know it by a different name.)

And can you evaluate the cube directly, in order to see what the answer will be, and perhaps get an idea for what they want you to do?

The more you tell us about what you know, and what has been said about the problem, the quicker we can see how to help. At the very least, what course are you taking, and is this your first exposure to complex numbers?

 

[reggiwilliams]

Have covered polar form
Did Complex Numbers several months ago and didn’t grasp it great then
Have looked back through my text book and if you think I need to evaluate (1+i(sqrt of 3)^3 in polar form can’t find any exercises where similar type question is asked
Can you steer me in right direction please

 

[Dr.Peterson]

Have covered polar form
Did Complex Numbers several months ago and didn’t grasp it great then
Have looked back through my text book and if you think I need to evaluate (1+i(sqrt of 3)^3 in polar form can’t find any exercises where similar type question is asked
Can you steer me in right direction please

You learned De Moivre's theorem; that should be sufficient. I'd expect you to have cubed complex numbers there.

What is the argument of the given number?

(By the way, don't look for similar questions; look for relevant ideas. Learning means being able to apply known ideas to new problems!)

 

[reggiwilliams]

Will have a study under this heading and get back to you tomorrow
Thanks for your help so far.

 

[reggiwilliams]

Starting to recall what I learnt previously
Lot of tricky work involved but have given it my best shot and just hope I’ve got it right.
Please let me know
Will need to do a lot of practice on this topic before I grasp it fully

 

[blamocur]

Looks good to me.

 

[reggiwilliams]

You learned De Moivre's theorem; that should be sufficient. I'd expect you to have cubed complex numbers there.

What is the argument of the given number?

(By the way, don't look for similar questions; look for relevant ideas. Learning means being able to apply known ideas to new problems!)

Hopefully I have this right now

 

[Dr.Peterson]

Hopefully I have this right now

Yes, that's correct.

Of course, you can also just multiply directly, though that doesn't appear to be what they are asking for, unless they just wanted you to conform that the answer has the right modulus:
[math]\left(1+i\sqrt{3}\right)^3=\left(1+i\sqrt{3}\right)\left(1+i\sqrt{3}\right)\left(1+i\sqrt{3}\right)\\=\left(1+i\sqrt{3}\right)\left(1+2i\sqrt{3}+i^2\sqrt{3}^2\right)\\=\left(1+i\sqrt{3}\right)\left(-2+2i\sqrt{3}\right)\\=-2+2i\sqrt{3}-2i\sqrt{3}+2i^2\sqrt{3}^2=-2-6=-8[/math]

 

[reggiwilliams]

Yes, that's correct.

Of course, you can also just multiply directly, though that doesn't appear to be what they are asking for, unless they just wanted you to conform that the answer has the right modulus:
[math]\left(1+i\sqrt{3}\right)^3=\left(1+i\sqrt{3}\right)\left(1+i\sqrt{3}\right)\left(1+i\sqrt{3}\right)\\=\left(1+i\sqrt{3}\right)\left(1+2i\sqrt{3}+i^2\sqrt{3}^2\right)\\=\left(1+i\sqrt{3}\right)\left(-2+2i\sqrt{3}\right)\\=-2+2i\sqrt{3}-2i\sqrt{3}+2i^2\sqrt{3}^2=-2-6=-8[/math]

Thanks again for your help