CPhill

Patterns for 2

2^1  = 2

2^2 = 4

2^3 = 8

2^4  = 16   and repeats   so    857 mod 4   =  1   so  2^857  ends in "2"

Patterns for 3

3^1  = 3

3^2  = 9

3^3  = 27

3^4  = 81    and repeats  so  927 mod 4 = 3   so  3&927 ends in "7"

So   2^857+3^927  ends in    2 + 7   = 9

And  2^857*3^927  ends in (2) (7)  = 4

This answer has  10,243  decimal digits  !!!  [ http://www.wolframalpha.com/input/?i=1024%5E1024*(-49)%5E7035*(-625)%5E2561%2F(-343%5E4690)

In other words.......I wouldn't worry about it.....!!!!

P  = 300 (2)^(D/11)

D          P

0          300

11        600

22       1200

33       2400

44       4800

etc.

http://www.wolframalpha.com/input/?i=300+(2)%5E(D%2F11)++for+D+%3D+0,+11,+22,+33,44

We can't factor this  in a "normal"  way  because it has no real roots

The facorizaion  is   [ x -  (- 1 -  i √  7  ) / 2 ]  [  x - (- 1 +  i √  7  ) / 2 ]

Expanding the above we get   →   x^2 + x + 2  ....exactly what we started wih  ....!!!!

[4/3](3.14)(70^3)  ≈  1,436,026.67  units^3

sin 60 cos 45 + cos 30 sin 30

[√3/2] [√2 / 2]  + [√3/2] [1/2]  =

( √3  [ √2 +  1 ] )  / 4  ≈ 1.04539

x^2 + x + 2.= 0    use the quad formula    a = 1, b=  1   c = 2

So we have

x  =  (- 1 ± √ [ (1)^2  - 4(1) (2)  ]  ) / 2  = 

(- 1 ± √ [ (1)  - 8  ]  ) / 2  =

(- 1 ±  i √  7  ) / 2

Key in  →      ( 50,031,545,098,999,707) ^ (1 / 35)

[ You should get "3" as your answer  ]

(64/729)^x = 243/32

(64/ 729)^x  =  3^5  / 2^5

( 2^6 / 3^6 )^x  = ( 2^5 / 3^5) ^(-1)

2^(6x)  / 3^(6x)  =  2^(-5) / 3^(-5)

(2/3)^(6x)  = (2/3)^(-5)    →  6x  = -5   →  x  = - 5 / 6

Let   < YTU  = x

Let   < XUT  = y

Let   < TVU  = z

< YIT  = <XIU  = 71

And we have the following system

(1/2) x + (1/2)y  + 109  = 180    →   x + y  =  142

x + y + z  = 180

(1/2)x + 71  + (1/2)y + 71 + 109 + z  = 360  →    x + y + 2z  = 218

So....

x + y  =  142      (1)

x + y + z  = 180   (2)

x + y + 2z  = 218  (3)

Subtract  (1)  from (2)  and z = 38° = < TVU  = "angle V "

And.....  y  = 142 – x  →   (1/2)y  = 71 - (1/2)x

As Melody  always says, "This is what I think. "