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By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4).  Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i.  Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

If we assume that sugar is measued in gallons too

water makes up 16 out of (16+ 3 + 1 = 20)  , or   16/20 ths of the final product

3 gallons  *  16 / 20 = 48/20 =  2 2/5   or   12 / 5 gallons

60 minutes to a full rotation of 360 degrees for the minute hand  = 6 degrees per minute

   13 minutes x 6 degrees/min = 78 ^o

   first letter consonant    21     x    5

       second letter consonant    5   x   21

           both letters different  vowels (different or alike)           5 x 5

105   + 105    + 25 = 235 'words' with at LEAST one vowel    

For   a  +  b =   n

The number of ordered-pair solutions  =   n  - 1

So   .......30  -  1  =     29

To see this.....consider  something  more  simple

Let   a  + b =  10

Solutions  are

1   9

2   8

3   7

4   6

5   5

6   4

7   3

8   2

9   1

The   number of  of solutions   =  10   - 1  =    9

4x - 2y = 8     re-arrnge to

y = 2x -4        slope , m = 2    parallel slope is the same   m  = 2

point slope form of new line through 0, -8

   y - - 8 =  2 (x -0)

   y = 2x - 8                          ( could have just changed the y intercept to -8   to get this result with a little less calc)

If  3  + 2i  is  a root  then  so is  3  -2i

-b  =  the sum of the  roots =  6

So  b  =  - 6

And  c  = the product of the  roots  =  (3 + 2i) ( 3 - 2i)  =  9 - 4i^2  =   9 - 4 (-1)  =  13

So the polynoimial is

x^2 - 6x  + 13

Wow I never thought of doing it that way! Thanks CPhill! 

Since  A  is reflected  over p  to B.....let the point of reflection on p  be = M

So angle AMX  = 90°

Likewise......since A  is reflected  over q  to  C....let  the point of reflection on q =  N

And angle  ANX  =  90°

So   angle  BAC  will =  angle  MAN

So

BAC  =  360 -  angle BXC - angle AMX  - angle  ANX  =   360 - 39 - 90 - 90   = 141°

No, the triangle gets larger but it's turned upside-down because of a negative scale factor. 

I also think that A is a hexagon, but I don't know what to do with this information. 

Here's a similar question: 

https://web2.0calc.com/questions/help_54234#r3

That's gonna look like this:

I think it might be 141, but I am not completely sure.

Future amount = $ 60 ( 1 + .05)³

Z' = -2, 3

Let  the  rate of one mechanic  = x

Let  the other's rate=   145 - x

So  we  have

20x   +  5 ( 145 - x)    =  2150      simplify

20x  +  725  - 5x  =  2150

15x + 725  =  2150

15x  =  2150   -  725

15x =  1425      divide both  sides  by  15

x =  1425   /   15    =  $95   

145  -  95   =   $50

The  two  rates  were  $ 95    and  $ 50

I'm pretty sure that the answer can be simplified to [A'B'C']/[ABC]. But from here I'm stuck...

I do not understand what a dilation of -4/3 is... does the triangle get smaller?

Yes, thanks! The answer was 10 

How did you get those answers though

I figured it out, sorry I am not sure how to delete this post.

$z^4=-4$

$z^4+4=0$

solving quartic polynomial, the roots are

$-1-i$

$-1+i$

$1-i$

$1+i$

Hi guest, your latex has not been shown, please edit and rewrite the questions without the latex or no one can solve. Sorry

That's pretty neat, heureka!!!!

I like this problem, even though I would have no clue as to the  solution!!!!

See here  :

https://web2.0calc.com/questions/evaluate-these-pls