Question

Can I get some help fan and medal....

Rationalize the denominator of square root of negative four over open parentheses 7 minus 3 I close parentheses plus open parentheses 2 plus 5 i.

quantity of negative 16 plus 18 I over 7

quantity of 16 plus 18 I over 145

quantity of 4 plus 18 I over 85

quantity of negative 4 plus 18 I over 77

Answer 1

The best way to get help is by showing your won work, not by spamming with multiple tags.

Give it a go! Let's see what you get.

Answer 2

I have literally no idea lol, I know it's something to do with conjugates though

Answer 3

:/

Answer 4

You simply can't have "no idea". Did you sleep through class? Do you have a book? Are you in the wrong class?

To "rationalize" is to find the right magic number that will get rid of everything that isn't rational.

\(\sqrt{2}\) Well, you need another \(\sqrt{2}\)

\(1 + \sqrt{3}\) Well, you need the conjugate, \(1 - \sqrt{3}\), takign advantage of the idea that (a+b)(a-b) = a^2 - b^2

\(3 + 2i\) Well, you need the complex conjugate \(3 - 2i\), for the same reason as the previous example.

Rationalize the denominator of square root of negative four over open parentheses 7 minus 3 I close parentheses plus open parentheses 2 plus 5 i.

Is it this? \(\dfrac{\sqrt{-4}}{(7-3i)+(2+5i)}\)?

Answer 5

That is the equation yes, now can u walk me through step by step I read the lesson Online school, but it wasn't very detailed only 1 page long

Answer 6

Just add the stuff in the denominator. Add the like terms and simplify it.

Answer 7

So the denominator becomes 9+2i?

Answer 8

Yes. Now, supply the Complex Conjugate of that number.

Answer 9

9-2i?

Answer 10

Perfect. Multiply both numerator and denominator by that value.

Answer 11

thats where i have trouble @tkhunny

Answer 12

So? Do it!!! Did you understand this part:

Taking advantage of the idea that (a+b)(a-b) = a^2 - b^2

That's all we're doing.

( 9+2i)( 9 -2i) = ??? Multiply away, just like any other binomial.

Answer 13

-4+18i? @tkhunny

Answer 14

What? I know you have multiplied binomials, before.

( 9+2i)( 9 -2i) = 91 + 18i - 18i - 4i^2

Ring any bells?

Answer 15

yeah I got that, then what about simplifying it?

Answer 16

You tell me. Add the like terms and watch the magic.

Answer 17

yeah I get 91-4i2 (squared) but thats not in the choices for denominator or numerator

Answer 18

Whoops! That's 81. Sorry for the typo.

Okay, now what was the point of the Complex Conjugate? Were we just making rice pudding or did we have a purpose?

Answer 19

Seems like rice pudding still :/

Answer 20

It may SEEM like that. But, what was our purpose? We wanted to accomplish something. What was it?

Answer 21

Weren't we gonna multiply by numerator and denominator?

Answer 22

Well, okay, that was the method, but that was not the purpose. We wanted to clear up the denominator and get all the Imaginary stuff out of there. Do you recall this purpose?

Answer 23

Sorry I am just so lost on this part of the lesson, I recall the purpose now.

Answer 24

We have 81 - 4i^2

Does that look Real and Rational?

Answer 25

yes

Answer 26

?? What's that "i" doing in there? That's not Real, is it?

Answer 27

oh wait woops that's imaginary so how do we get rid of it?

Answer 28

i^2 = What???

Answer 29

-i? or -1?

Answer 30

\(\sqrt{-1} = i\) therefore, \(i^{2} = -1\)

81 - 4i^2 = 81 - 4(-1) = 81 + 4 = 85

Wow!! Anyone remember where this all started?

\(\dfrac{\sqrt{-4}}{(7-3i)+(2+5i)} = \dfrac{i\sqrt{4}}{(9+2i)} = \dfrac{2i}{(9+2i)}\)

That was just simplifying.

Then, we introduced the Complex Conjugate.

\(\dfrac{2i}{(9+2i)}\cdot\dfrac{9-2i}{9-2i} = \dfrac{18i - 4i^{2}}{81-4i^{2}} = \dfrac{18i + 4}{81+4} = \dfrac{4+18i}{85}\)

It wasn't nearly as bad all in one place without all the discussion and missteps. Follow each step and make sure you know where it came from.

Answer 31

okk thanks, I was lost because the first step after "then we introduced the complex conjugate" I didn't have and such a mess, but I totally understand, thanks so much @tkhunny

Answer 32

You really hung in there. Good work!

Now, go do 100 more! :-)

Answer 33

lol, I'll try.