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what if it was a rational number like -5.2 or something, would that still be an integer? 

oh and thank u for the help! :)

appears to be 15049

pi / 0    is not a defined quantity

Negative rationals CAN be integers  [ Ex :   -24/6 = -4  ]

Irrationals can never be integers  [ positive or negative ]

\(z+\dfrac 1 z = z^2 + \dfrac{1}{z^2}\\ z^4-z^3-z+1=0\)

\(\text{A quick eyeball solution is }z=1\\ \text{So let's divide through by }(z-1)\\ z^3-1 = 0\\ \text{This also clearly has a solution at }z=1 \text{ so divide again}\\ z^2+z+1 = 0\\ \text{This is just quadratic so we can apply the quadratic formula}\\ z = \dfrac{-1\pm \sqrt{1-4}}{2} = -\dfrac 1 2 \pm i\sqrt{3}\\ \text{So in total we have}\\ z = 1, ~z= -\dfrac 1 2 + \dfrac{i\sqrt{3}}{2}, ~z = -\dfrac 1 2 - \dfrac{i\sqrt{3}}{2}\)

\(w = 1 + \dfrac 1 1 = 2\\ -\dfrac 1 2 \pm i \sqrt{3} = e^{\pm i\frac{2\pi}{3}}\\ e^{\pm i\frac{2\pi}{3}}+e^{\mp i\frac{2\pi}{3}} = 2\cos\left(\dfrac{2\pi}{3}\right) = -1 \)

ill make sure to def take note of this, thank you SO much, sending virtual fist bumps! lol

Don't know much about this.....but.....

The correlation coefficient is  ≈ .97    which is very high....thus...the regression line is a very good fit

Predicted days  are found by  plugging  the  vacation days available into the regression formula for x

The Residual Column  =  actual days - predicted days

So.....

Predicted Days          Residual

7.2                             -1.2

15.6                           -1.6

6                                  2

24                                1

20.4                            -1.4

10.8                             0.2

10.8                            -1.8

20.4                             0.6

8.4                              -0.4 

13.2                             2.8

Here is the residual graph....the predicted point is plotted first with its residual point second

https://www.desmos.com/calculator/hzkkows7m0

The center of the ellipse will be the origin

The length of the major axis is 4....so a is 1/2 of this = 2 

The length of the minor axis = 2sqrt(2)....so.....b is 1/2 of this = sqrt(2)

So....the equation of the ellipse is

x^2            y^2

___    +      _________         =    1

 2^2            [sqrt(2)]^2

x^2          y^2

___ +   _____    =     1

  4            2

The area of the ellipse   =  pi * a * b  =  pi * 2 * sqrt(2)  =   2sqrt(2) pi

So

k  =  2sqrt (2)

Here is the graph : https://www.desmos.com/calculator/ovso9le6pj

We would have to take into account the parts of the universe that it was alone in. I wouldn't know if it is moving because it was the only thing in the universe.

Can't tell what the letter at the top left of the figure is...but...I think it is "H"

Anyway....join  H and Y 

Triangle HAY   is a isoceles right triangle.....with HX = XY

So....

HY =  sqrt [ HX^2 + XY^2]  =  sqrt [ 4^2 + 4^2 ] =  sqrt [ 32]   

And......HY will create  triangle HYZ

Angle HYX =  135 - Angle  HYX

Angle HYX = 135 - 45  = 90

So....triangle HYZ is right wth HZ the hypotenuse  and HY one of the legs

So

"a"  = ZY  =  sqrt [ HZ^2 - HY^2]  =  sqrt [ 9^2 - 32]  = sqrt [ 81 - 32] = sqrt  [49]  =  7

17   37 47  67 97

LOL!!!.....don't do that.....I'm out of Kleenex  !!!!

1   6!   ways to list pres candidates   x  4! ways to list VPs  x  5!   to list Sec candidates x 3! for treas

    that is   6!  x 4! x 5! x 3!    =...........    (You can do the number crunch...HINT:I get 12 million -something)

2   Less than 500 means only 4 choices for the first position   x   5  choices for second position x 5 choices for third position=

      4 x 5 x 5 = 100 possible combos total    if it ends in 2 or 4 it will be even    2or 4 out of 5  is 2/5    2/5 x 100 = 40 possibilities for even

3  3 ways to choose lineman      

        leaves 9 people to fill 3 positions         3   x   9c3 = 252 posssibilities

4     9 p 4  = 3024    ( I think  )

( n + 1)^3  

  _______   simplifies  to       n^2   - 17n - 6869   +    343

    n + 20                                                    _____

                                                                   n + 20

6859   factors as 19^3

So....19^2   will divide 6859......so   

n + 20  = 19^2 

n = 19^2 - 20  =  341

Also....19^3 will divide 6859....so

n + 20  =  19^3

n =  19^3 - 20   = 6839

So.....the two positive integers are   n = 341    and n = 6839

wow this was waay far from what i was doing this is so helpful, thank you so much, i could cry ;-;

Work is using a force to move an object a distance (when both the force and the motion of the object are in the same direction).

The answer would be c), since you are using a force to move the baseball bat a distance.

/ᐠ｡ꞈ｡ᐟ\ /ᐠ｡ꞈ｡ᐟ\ /ᐠ｡ꞈ｡ᐟ\

Percentages and factors are deceiving......   If there was only 1 or maybe 3  surfboard rentals last year......doubling that would still only be 2 or 6...not very many surf board rentals.

Maybe as the temperature goes up....or as the days get longer (in the summer)  ...or there are more tourists during a particular season, the popsicle consumption and surfboard rentals both go up !

I think I can do some

I'm letting x  =  females  and y = males

We have the points    

(64, 82)   (82, 77)  (70, 65)  (59, 58)  (82, 76)  ( 56, 61)  (81, 76)  (63, 57) ( 74, 70)  (80, 76)

Using this website.....https://keisan.casio.com/exec/system/14059929550941

The linear regression equation is

y = .6134x + 26.187

The correlation coefficient  = r  ≈  .692    [moderate correlation ]

The coefficient of determination  = r^2 ≈  .479

Here is the scatterplot :

It appears that we have at least one outlier  ....(64, 82)

A  cluster appear to be the points  (80, 76)  (81, 76)  ( 82, 76)  ( 82, 77)  

Hope that helps some.....!!!!

First one :

B. All positive real numbers

However......if miles can only be (rounded) integer values and 0 miles is possible, then the answer might be

A. All whole numbers

Second one :

A.  All  whole numbers  (we can have 0 people waiting  or a positive integer  number of people waiting )

Find log 2         then find   log 3       then      the average is   (log 2 - log 3)/(2-3)

Interesting metaphors.  Generating functions seemed magical to me the first time I saw them demonstrated.  Along with usable numbers popping up like dandelions in May, I actually understood the counting methods used for its derivation, without having to spend hours studying. That was magical for sure.  My mentor, Lancelot Link, who taught the concepts to me wasn’t completely human, and he certainly was from high-up and far-out, but not quite to outer space –though I still do wonder sometimes.