Writing expression in form 1+bi? easy 10 points.?

Question

Write each expression in the form a + bi.

1. 1 / (2+5i)

2. (5+i) / (5-i)

3. (3+i sqrt(2)) / (7-i sqrt(2))

Thank you!

Answer 1

multiply everything by its conjugate

1. 1 / (2+5i)

    1 ( 2 - 5i)  / (2+5i)(2-5i)

     (2-5i) / (4 -25i^2)

     (2-5i) / (4+25)

     (2-5i) / 29

2. (5+i) / (5-i)

    (5+i)(5+i) / (5-i)(5+i)

    (25 + 10i + i^2) / (25 -i^2)

    (25 + 10i - 1) / (25 + 1)

    (24+10i) / 26

    reduce everything by two

    (12 + 5i) / 13

3. (3+i sqrt(2)) / (7-i sqrt(2))

    (3+ isqrt2)(7+ isqrt2) / (7 - isqrt 2)(7 + isqrt 2)

    (21 + 3isqrt2 + 7isqrt2 + [isqrt2]^2)  / (49 - [isqrt2]^2)

    (21 + 10isqrt2 -2) / (49 + 2)

    (19 + 10isqrt2) / 51

Answer 2

1/(2 + 5i) * (2 - 5i)/(2 -5i)= ( 2 - 5i)/(4 +25) = (2 - 5i)/29= 2/29 - 5i/29

#2

(5+i) / (5-i) * (5 + i)/(5 + l)= (25 + 10i + i^2)/(25 - i^2)= 24 + 10i/26 =

24/26 = 10I/26-----. 12/13 + 5i/13

#3

`19/51 + 10isqrt(2)/51

Answer 3

1. 1 / (2+5i)=(2-5i)/[(2+5i)(2-5i)]=(2-5i)/(4+2... - 5/29 i

2. (5+i) / (5-i)=(5+i)^2/[(5-i)(5+i)]=(25+10i-1)/(2... + 10/26 i=

=12/13 + 5/13 i

3. (3+isqrt(2))/(7-isqrt(2))=(3+isqrt(2))(7... (21 + i 7 sqrt(2) + i 3 sqrt(2) -2)/(49+2)=19/51 + 10 sqrt(2)/51 i

I hope you understand and good luck!

Answer 4

Question 1

( 2 - 5i )

----------------------

( 2 + 5i ) ( 2 - 5i )

( 2 - 5i )

-----------------------

4 - 25 i ²

2 - 5 i

---------

29

Question 2

( 5 + i ) ( 5 + i )

-----------------------

( 5 - i ) ( 5 + i )

25 + 10 i  - 1

---------------------

25 - i ²

24 + 10 i

-------------

26

12 + 5 i

------------

13

Question 3

( 3 + i √2 ) (7 + i √2)

--------------------------------

( 7 - i √ 2 )( 7 + i √ 2)

21 + i 10 √2 - 2

--------------------------

49 - 2 i²

19 + i 10 √2

------------------

51