CPhill

4 + 1/4 =  17/4

4 + 4/17  =  72/17

4 + 17/72  = 305/72

1 + 72 / 305  =  377/305

1/ [ 377/305] =   305 / 377

\(\frac{1}{\sqrt{36} + \sqrt{39}} + \frac{1}{\sqrt{42} +\sqrt{45}} + \frac{1}{\sqrt{45} + \sqrt{49}} \)

[ sqrt 36 - sqrt 39] / [ 36 -39 ]    +  [ sqrt 42  -sqrt 45 ] / [42 - 45]  +  [ sqrt 45 - sqrt 49] / [45 - 49 ]  =

[   6 - sqrt 39 ] / -3     +  [ sqrt 42 - sqrt 45 ] / -3   +  [ sqrt 45 - 7 ] / - 4  =

[ sqrt 39 - 6 ] / 3  + [ sqrt 45 - sqrt 42 ] / 3   + [7  - sqrt 45 ] / 4

[ 4 sqrt 39 - 24 + 4sqrt 45 - 4sqrt 42 + 21 - 3sqrt 45 ]  /   12

[ -3  + sqrt 45  + 4sqrt 39  - 4sqrt 42 ] / 12  =

[  -3 + 3sqrt 5  + 4sqrt 39 -  4sqrt 42 ]  / 12

Identifiable sequences  = 

7!  / 2!  =     2520

                         20 A

                                      E                    

                      144 H        

        B                  D               C

ABH forms a triangle

Angle AHB = 144

Angle BAH = 20

So angle ABH =  180 - 144 - 20  =  16° 

Triangle HQN similar to Triangle FQG

HN / FG =  2 / 7

Therefore  HQ / FQ  = 2 / 7

Therefore  HQ / (HQ + FQ) =  HQ / HF  =  2 / ( 2 + 7)  = 2/9  ⇒ HQ = (2/9) HF

Triangle PFM similar to Triangle PHG

FM / HG =  3 / 8

Therefore PF / PH = 3 / 8

Therfore PF / (PF + PH)  = PF / HF  = 3 /( 3 + 8) = 3 / 11 ⇒  PF = (3/11) HF 

Getting a common denominator  between  9  and 11

HQ = 22/99FH

PF  = 27/99 FH

Therefore

FH - HQ - PF  =  PQ

FH - (22/99)FH - (27/99)FH  = PQ

(50/99)FH  = PQ

Therefore

PQ / FH  =  50 / 99

See this

AE = EB  =  2.5

Height of EAD = Height of EBF =  4

Area of each triangle =  (1/2) ( 4) (2.5)  =  5

Triangle FDC  has a base and  height twice that of EAD.....so....its area = 2^2 * 5   = 20

[ FDC ]  - [ EBF]   =  20  -  5 =   15   = area of EBCD

[ EAD ] +  [ EBCD ]  =  5 + 15  =    20    =   [ ABCD  ]

Simplify as

21mn - 2m - 8n + 8

Add  -19m

So we have

21mn - 21m - 8n + 8  =

21m ( n - 1)  -  8 (n - 1)  =

(21m - 8) ( n - 1)

                      X                          Y

                      90                     120

      60

      W                             Z

XZW  will form a triangle

Angle ZXW  = 90

Angle XWZ = 60

So XZW =  180 - 90 - 60  =   30

Simplify  as

(7 + sqrt 5 + 2sqrt 3)^2 =    (7 + sqrt (5) + sqrt (12))^2

[ (7 + sqrt 5) + sqrt (12) ]  [ (7 + sqrt (5) + sqrt (12) ] =

(7 + sqrt 5)^2 + 2 [ (7 + sqrt (5)) * sqrt (12) ]  +  (sqrt (12))^2  =

49 + 14 sqrt 5 + 5  + 2*7 sqrt (12) + 2sqrt (60) + 12  =

66 + 14sqrt (5) + 14 sqrt (12) + 4sqrt (15)  =

66 + 14sqrt (5) + 28sqrt (3) + 4sqrt(15)

Simplify  as  ( 3y - 11)^4

Expansion

C(4,0)*(3y)^4 =   1* 81 * y^4  =   81 y^4