Question

write the complex number in standard form:

Answer 1

\[\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right) ^4\]

Answer 2

@saifoo.khan

Answer 3

Since I'm lazy, the first thing that came to mind was use the binomial formula:\[(x+y)^n=\sum_{k=0}^{n}\binom nkx^{n-k}y^k\]

Answer 4

^i havent learned that yet and im very sure that's not what im supposed to use

Answer 5

@experimentX

Answer 6

the answer is in the back of my book and it says that it's -1 and my calculator gives the same answer but i dont understand HOW??

Answer 7

@lgbasallote

Answer 8

anyone wanna take a whack at it...kinda need this :\

Answer 9

sure!

Answer 10

Using what I told you,\[\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\right)^4=\sum_{k=0}^{4}\binom4k\left(\frac{\sqrt{2}}{2}\right)^{4-k}\left(\frac{\sqrt{2}}{2}i\right)^k=\frac14+\frac14+i-\frac32-i=-1\]

Answer 11

Recall that\[\binom ab=\frac{a!}{b!(a-b)!}\]

Answer 12

again...I. have. absolutely. no. idea. what. that. is.
comprehend? :)

Answer 13

On the other hand, you can just expand\[\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right)\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right)\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right)\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right)\]and simplify.

Answer 14

\[(\sqrt2/2+\sqrt2/2i)^4=((\sqrt2/2+\sqrt2/2*i)^2)^2\]

Answer 15

\[\left({2\over4}+{2\over4}~(-1)\right)^2\]

Answer 16

\[[(\sqrt2/2+\sqrt2/2*i)(\sqrt2/2+\sqrt2/2*i)]^2\]

Answer 17

no you forgot the middle term

Answer 18

\[2/4+2/4i+2/4i+2/4(-1)\]

Answer 19

squared

Answer 20

\[\left({2\over4}+{2\sqrt{2}\over4}~~i+{2\over4}~(-1)\right)^2?\]

Answer 21

no the middle term is\[ 2(\sqrt2/2i)(\sqrt2/2) \]

Answer 22

or i

Answer 23

k forget this, im not understanding the tiniest bit of this and ive got more stuff to do...ill just ask my teacher tomorrow...thanks anyway @Fgcbear16! :)

Answer 24

ok. sorry

Answer 25

dont be...my fault....i appreciate your help! :)

Answer 26

how did you get your answers to look like that?

Answer 27

how did i get my what to look like what?

Answer 28

your responses were look different than mine in format.

Answer 29

oh i write our the equation forms....using the "Equation" button below..once you get to know it a little better you'll be able to write them without using that button ...explore and you'll find! ;)

Answer 30

how did you the upright fractions?

Answer 31

make use of De'Moive formula
\[ \left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right) ^4 = \left( \cos {\pi \over 4}+\sin {\pi \over 4}~i\right) ^4 = \cos \pi + i \sin \pi = -1\]

Answer 32

alright guys i got this :DD

lego!!:
\[\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}~i\right) ^4\] \[\left[\left(\frac{\sqrt{2}}{2}\right)\times ~( 1+i)\right] ^4\] \[\left[\left(\frac{\sqrt{2}}{2}\right)^2\right]^2\times~ \left[( 1+i)^2\right]^2\] \[\left(\frac{2}{4}\right)^2\times ~\left( 1+i\right)^4\] \[{4\over16}\times\left(1+i\right)^4\] \[{1\over4}\times-4\] \[\large =-1\]

\[\huge OH~~~ YEAH! ~~ :)\]