CPhill

\((\sqrt[6]{4})^9 \)   =

[ 4^(1/6) ] ^9   =

4^(9/6)  =

4^(3/2)  =

(4^3)^(1/2)  =

64^(1/2) =

√64  =

8

Deleted......

3y + 2a = 4y +7 - y +3       simpify

3y + 2a  = 3y + 10

This will have infinite soutions when the left side  = the right side....so this implies that.

2a  = 10     divide both sides by 2

a  = 5

Ah....you a referring  to the concept of the "Riemann Sum"

Here is a very good tutorial on this....it explains it much more clearly than I can :

http://tutorial.math.lamar.edu/Classes/CalcI/AreaProblem.aspx

I'm not sure, but I think this might be something known as the "average value"

For instance......let us find the area of the curve between the function  1 - x^2   and the x axis

The curve will intercept the x axis at   x = - 1    and x  = 1

So.....using symmetry, the area is

    1                                                      1

2* ∫  1 - x^2  dx     =    2  [ x - (1/3)x^3 ]       =   2 [  1 - 1/3]  =  2 [ 2/3] =  4/3  units^2

    0                                                      0

Note that we can construct a rectangle with the same area if we let the width range   from x  = - 1 to x  = 1  =  2 units = the interval width

So....the height of this rectangle  must be      (4/3) / 2  =  4/6  = 2/3

See this image :

Note that the rectangle ABCD  has the same area as the definite integral

To   always find this average value  we can calculate this :

[ area  of integration ]

_________________    =      height of rectangle  =  average value

  interval width

You didn't miss anything

F(x)  = y

So

F(-2)  =  y  ???

So    (x, y)  = (-2, y)    and the point with an x coordinate of -2   has a y coordinate of  2

So

f(-2)  = 2

F(-2)   actually asks..."When we put "-2" into "F".....what is the resulting y value???"

Look  at the points (x , y)....note that we have the point  (-2, 2)

This means that when we put -2  into F, we get get 2 as a result

So

F(-2)  =  2

1 fraction with 2 as a denominator

2 fractions with 3 as a denominator

3 fractions with 4 as a denominator

.........

n fractions with n + 1 as a denominator

Note that  (3)(4)/2  = 12/2  =   the 6th term = 3/4

Also note that (13) (14)/2  =  182/2  = the 91st term = 13/14

So......the 99th term =   1/15, 2/15, 3/15, 4/15, 5/15, 6/15, 7/15, 8/15

101     132   156   187       202     231    257    286

110     134   165   189       211     235     264    297

112     143   167   198       213     242     268

121     145   176                220     246     275                 

123     154   178                224     253     279   

Total = 35

3x^2 + nx  + 72

The greatest pairwise sum of the factors of 72 is....  72  and 1   = 73....so....we can write

(3x + 1) (x + 72)   =   3x^2 + 1x + 216x + 72  =  3x^2 + 217x + 72

So....the largest value of n  =  217