CPhill

2 hrs 40 min =  2 + 2/3  hrs =  7/3 hrs

3 hrs 30min = 3 + 1/2 hrs  =  7/2 hrs

Distance =   Rate * Time

Let  the speed of the plane in still air =  S

When there is no wind it is traveling  (S)  mph    and against the wind it is traveling (S - 50) mph

So.....equating the same distance

S * ( 7/3)  =  (S - 50) ( 7/2)

(7/3)S  = (7/2)S -  175

175 = ( 7/2 - 7/3)S

175  =  (7/6)S

S = 175 ( 6/7)  =    150 mph in still air

So....the distance between cities  is   (150) (7/3)  =  350  miles

\( \dfrac{1}{\sqrt2+sqrt3}+\dfrac{1}{\sqrt2-sqrt3} \)

1 [ sqrt2 - sqrt 3]   +   1 [sqrt2 + sqrt 3]

_________________________________  =

  (sqrt 2 + sqrt 3) ( sqrt 2 - sqrt 3)

2sqrt 2

______  =

2 - 3

-2 sqrt 2

\( \frac{4}{1 - \sqrt[3]{3}} + \sqrt[3]{9} \)

Note that    1 - 3    can be factored as a difference of  cubes  =   ( 1 - ∛3) ( 1 + ∛3 + ∛9)

Multiplting the denominator of the first fraction by  ( 1 + ∛3 + ∛9)  we have

[ 4  ( 1 + ∛3 + ∛9) ] /  [ 1 - 3 ]   +  ∛9   =

[ -2 (  ( 1 + ∛3 +∛9) ] + ∛9  =

-2 - 2 ∛3 - 3∛9

2ab -12a + 3a + 3b - 8b - ab       simplify

ab -9a - 5b

a ( b - 9)  - 5 ( b - 9)   =

ab  - 9a  - 5b  + 45

mn = 3m + 6n -2m + 4n

mn = m + 10n

mn - m  = 10n

m ( n - 1)   = 10n

m =  10n  / [ n - 1]

m =  10  [ n]/[n -1]

m        n

20       2

15       3

12       6

11      11

 0        0

-1        5

-4        8

-9        9

 (1-3i)(2-i)(2+i-7+5i)(1+3i-2+4i)

(1 -3i) (2-i) (-5 + 6i) (-1 + 7i)

[ 2 - 6i -i +3i^2 ] [ 5 - 6i  -35i + 42i^2]

[ -1 - 7i ] [ -37 - 41i]  =

[ 37 + 259i + 41ii + 287i^2]  =

[ 37 + 300i - 287 ]  =

[ -250 + 300i]

[15 - 16i]  / [ 2 -3i + 5 -7i]  =

[15 -16i] / [ 7 -10i]    =   

[ 15 -16i] [ 7 + 10i]

_______________   =

[ 7 -10i] [ 7 +10i]

[ 105 - 112i + 150i - 160i^2 ]

_______________________  =

49 - 100i^2

[ 105 + 160 + 38i ]

------------------------  =

49 + 100

[ 265 + 38i ]

__________

     149

a11 = a10 + a9

1 = 4  + a9

a9 = -3

a10 = a9 + a8

4 = -3  + a8

a8 = 7

a9  = a8 + a7

-3 = 7 + a7

a7 =  -10

a8  = a7 + a6

7 = -10 + a6

a6 = 17

By inspection  ( a,b,c,d,e) =  (1,0,0,0,0)

A first letter  =  15 ways

E first letter = 11 ways

I first letter = 7 ways

O first letter = 1 way

So  34 total ways