## EarthCitizen 4 years ago ((3+j3)^3*(1-j)^4)/((1+j(3)^1/2)^9)

1. EarthCitizen

$((3+j3)^{3} *(1-j)^{4})/(1+j \sqrt{3})^{9}$

2. EarthCitizen

yo

3. asnaseer

Just digesting that equation :)

4. asnaseer

ok - it would be best to try and break it down a bit to simplify it first...

5. asnaseer

$(3+3j)^3=(3(1+j))^3=3^3(1+j)^3=27(1+j)^3$

6. asnaseer

next simplification...

7. asnaseer

$(3+3j)^3*(1-j)^4=27(1+j)^3*(1-j)^4=27*((1+j)(1-j))^3(1-j)$$=27*(1-j^2)^3(1-j)=27*(1+1)^3(1-j)=27*2^3(1-j)$$=27*8(1-j)=216(1-j)$

8. asnaseer

make sense so far?

9. EarthCitizen

yep

10. asnaseer

ok, so now we are left with:$((3+j3)^{3} *(1-j)^{4})/(1+j \sqrt{3})^{9}=\frac{216(1-j)}{(1+j\sqrt{3})^9}$

11. asnaseer

next lets try and simplify the denominator...

12. EarthCitizen

alryt

13. asnaseer

$(1+j\sqrt{3})^2=1+j2\sqrt{3}+3j^2=1+j2\sqrt{3}-3=-2+j2\sqrt{3}=-2(1-j\sqrt{3})$therefore:$(1+j\sqrt{3})^3=(1+j\sqrt{3})^2(1+j\sqrt{3})=-2(1-j\sqrt{3})(1+j\sqrt{3})$$=-2(1-j^23)=-2(1+3)=-2*4=-8$

14. asnaseer

therefore:$(1+j\sqrt{3})^9=((1+j\sqrt{3})^3)^3=(-8)^3=-512$

15. asnaseer

can you complete the rest now?

16. EarthCitizen

uhmm.., cpould you plz divide the top by the bottom for a final solution ?

17. asnaseer

$((3+j3)^{3} *(1-j)^{4})/(1+j \sqrt{3})^{9}=\frac{216(1-j)}{(1+j\sqrt{3})^9}=\frac{216(1-j)}{-512}=-\frac{27(1-j)}{64}$

18. EarthCitizen

19. asnaseer

yw - the main point however is that you understand the process - I hope you did :)

20. asnaseer

the "key" I guess is spotting how to get the equation into a form like: $$(a+jb)(a-jb)$$ so that it can be simplified to $$(a+jb)(a-jb)=a^2+b^2$$

21. asnaseer

that will come with practice.

22. EarthCitizen

yh, hang on. so you used difference of two squares only ?

23. asnaseer

sum of two squares

24. asnaseer

$a^2-b^2=(a+b)(a-b)$$a^2+b^2=(a+jb)(a-jb)$

25. EarthCitizen

yh, so why didn't you use de Moivre's theorem ?

26. asnaseer

you could, but then you would need to find the correct angle for:$e^{j\theta}=\cos(\theta)+j\sin(\theta)$and I /feel/ the approach I used instead is often much simpler.

27. EarthCitizen

yh. a lot simpler coz it started getting cloudy , thanx tho

28. asnaseer

np - you are more than welcome. I'm glad I was able to help.