CPhill

A = ( 4cos[ pi/4 ]   , 4 sin [ pi /4 ])     =  (4cos 45° , 4sin 45°)  =      (4 / sqrt( 2)  , 4/sqrt ( 2 )  )

B =  (3 cos [ 5pi/12 ],  3sin [5pi/12 ] )  =  (3cos 75° , 3sin 75°)  = 

(  3 [  sqrt (3) - 1 ] / [ sqrt (8)]  ,  3 [ sqrt (3) + 1] / [sqrt (8)]   )

C = ( 2, pi/12)  =  (2 cos 15° , 2sin 15°) =   (2 sin 75° , 2cos 75°) =  

( 2 [sqrt (3) + 1]/ [sqrt (8)]  ,  2[sqrt (3) -1 ]  / {sqrt (8)  ] 

Vinculum ...   Do you want   AB^2  , AC^2  and BC^2  ??

Or do you mean  (AB)^2 , (AC)^2  and  (BC)^2   ??

Also....these are real points......there shouldn't be any  "i"    involved

Note that  ∛16 =   ∛2^4  =  2 ∛2

So....the denominator just becomes

∛2 + 2∛2 =   3∛2

So    we  have

  2

___    multiply num / den   by     2^(2/3)  = ∛(2^2) =  ∛4

3∛2

2 * ∛ 4                2 * ∛4          2 * ∛4           ∛4             4^(1/3) /  3   

_______  =     _______  =  _______  =   _____ =   

3∛2 * ∛4            3 * ∛8           3 * 2              3

A +  B  =   4 + 3   =  7

x cannot be negative.....so......the domain  is  [0 , inf )

The pentagon will be composed of 5 congruent isosceles triangles

Their equal sides  = 12   and  the  included angle between  these sides =  72°

The  area of one of these =   (1/2) (12^2) sin (72°)  =   72sin 72°

So the area is ....5  (72)sin (72°)  =   360 sin (72°)   ≈    342.38   cm^2

x - y  =   36         (1)

4y - 3   =  x       (2)

Sub (2)  into (1)

(4y - 3 )  - y    =  36

3y - 3  =  36

3y  = 36 + 3

3y = 39

y =  39 / 3   =   13  =  smaller

And   4(13) - 3  =  x  =  52 - 3  =  49  = larger

A)   We are using the property     ^c √  (a^b)    =   a^(b/c)....so...

  ⁴ √ x^3 =    x^(3/4)

B)  We are using the property that   1 / a^-n  =   a^n .....so.....

1 / x^(-1)   =  x^1   

C)  We are using the property  that   a^m * a^n  =  a^(m+n)

¹⁰ √ (x^5  * x^4  * x^2 )  =    

¹⁰ √ (x^11)  =

x^(11/10)

D)  x^(1/3) * x^(1/3) * x^(1/3)   =

      x^(1/3 + 1/3 + 1/3)  =

     x^1 

\( $\frac{1+\sqrt{2}}{1+\sqrt{3}} \cdot 2$\)

\($-A-\sqrt{B}+\sqrt{C}+\sqrt{D}$\)

Multiply  num  /  den  by       1 - sqrt 3

(1 + sqrt 2) ( 1  -sqrt 3)          1 -sqrt 2 + sqrt 3 - sqrt 6          1  - sqrt 2 + sqrt 3 - sqrt 6

__________________ =      ____________________ =     _______________________  

( 1 + sqrt 3) (1 - sqrt 3)                  1 -  3                                                 -2

Multiplying this result by 2  we get

- 1 ( 1 - sqrt 2  + sqrt 3  - sqrt 6 )  =

-1 - sqrt 3 + sqrt 2  +  sqrt  6

A + B + C + D   =      1  + 3  + 2  + 6   =     12

Note that f o g    means we are putting  g  into f

1)    Let    f  =   sin x    .......  a sinusoidal  function is  continuous everywhere

      Let    g =  x^3 + 7x + 1......this is a polynomial....so it is continuous everywhere

So   f  o  g  =       sin (x^3 + 7x + 1)    is continuous

2)   Let   f =    lx l  .......the basic absolute value function is  contimuous everywhere

      Let  g =   sin x    (continuous  everywhere )

So    f o  g  =    l sin x  l    is continuous

3)    As before    

f  = cos ^3  x          sinusoidal function { continuous  everywhere }

g =  x +1                linear function  {continuous everywhere}

So

f o g  =    cos^3 ( x + 1)      is continuous

(1/2)x^2  +  99x  + c  =  0

We have the form     ax^2 + bx + c  = 0

The product of the roots =  c / a

So

[-99 -sqrt (9577) ]  * [ -99 + sqrt (9577) ]  =    c / (1/2)

9801 - 9577  =  c / (1/2)

224 = c / (1/2)

224 (1/2)  =   c   =    112

Let the measure of one interior angle = x

And its exterior angle = 180  - x

So

x / ( 180 - x)  = 2.5

x = 2.5 (180 - x)

x = 450 - 2.5x

x + 2.5x =  450

3.5x  =  450

x =  450 / 3.5  =  the measure of an exterior angle

The  number of sides is given by

(N - 2) (180) / N  =   450/ 3.5        simplify

180N - 360 =   (450 / 3.5) N

180N - (450/3.5) N  =  360

(630 - 450) N   =   360 * 3.5

180N  =  1260

N  = 1260 / 180  =    7   =  the number of sides