## anonymous one year ago Find the quotient. Write your answer in standard form. 3 + i / 3 - i A. -1 B. 1 - i C. 3/5 + 4/5 i D. 4/5 + 3/5 i

1. Nnesha

multiply top and bottom both by the conjugate of the denominator (3-i)

2. Nnesha

do you know the definition of conjugate ?

3. anonymous

no. ): @Nnesha

4. Nnesha

conjugate example just change the sign of imaginary number a+bi conjugate is a-bi bi imaginary a= real so 3-i conjugate is what ?

5. anonymous

3+i ?? @Nnesha

6. anonymous

is it $\frac{3+i}{3-i}$?

7. anonymous

if so, multiply top and bottom by the conjugate of the denominator the conjugate of $$a+bi$$ is $$a-bi$$ and this works because $(a+bi)(a-bi)=a^2+b^2$ a real number

8. anonymous

first step is $\frac{3+i}{3-1}\times \frac{3+i}{3+1}$ which gives you $\frac{(3+i)(3+i)}{3^2+1^2}$

9. anonymous

the denominator is evidently $$10$$ and the numerator is whatever you get when you multiply that out

10. anonymous

you good from there?

11. anonymous

sorry typo there i meant "first step is $\frac{3+i}{3-i}\times \frac{3+i}{3+i}$

12. anonymous

I got (9 + 6i + i^2) / (9 - i) I don't know where to go from there? @satellite73

13. anonymous

ok lets back up, your denominator is wrong

14. anonymous

Is it 9-i^2?

15. anonymous

it is not $$9-i$$ but rather $$9+1=10$$ $3-i$ has real part $$3$$ and imaginary part $$-1$$ when you multiply $(3+i)(3-i)$ you get $3^2+(-1)^2$ or $$10$$

16. anonymous

you can write it that way if you want, but don't forget $$i^2=-1$$ so $$9-i^2=9+1=10$$

17. anonymous

So is i by itself 1?

18. anonymous

you should really just be thinking that $(a+bi)(a-bi)=a^2+b^2$ just like in pythagoras

19. anonymous

no $$i\neq 1$$ but $$i^2=-1$$ and so $$-i^2=+1$$

20. anonymous

probably the confusing part is how to interpret $$a+bi$$ when you have $$3-i$$ in that case $$a=3,b=-1$$ of course when you square, you can ignore the minus sign

21. anonymous

your numerator of $9+6i+i^2$is correct, but not complete since $$i^2=-1$$ then $9+6i+i^2=9+6i-1=8+6i$

22. anonymous

giving you an almost final answer of $\frac{8+6i}{10}$ now cancel the common factor of 2 top and bottom

23. anonymous

So it's D!! I UNDERSTAND!! Thank you so much. (:

24. anonymous

yes it is D, and you are welcome

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