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As long as " a "  ≥ 0....then

The cube root can be written as a^(1/3)

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

So....the square root of a cube root is

(a^1/3)^(1/2)  = 

(a)^(1/2 * 1/3)  =

a ^(1/6)     ....  i.e.,  the sixth root

This only holds as long as "a" ≥ 0

If "a" <  0.....the  cube root of this is negative...then......  we are taking the square root of a negative.....which is not a real number

Does this make sense, GM ???

Yes she is correct because when you convert it into exponential value like so: \((a^{1/3})^{1/2}\) and then use the Product Property of Exponents you get \(a^{1/6}\) which is equivelent to the 6th root. Also lets say a is 64 and you take the cube root and get 4 and then take the square root of 4 and get 2. Also the 6th oot of 64 is 2. So this shows that they are both equal.

thank you !!

Notice that when x = -1, y = f(-1)  =  3

And when x = 1, y = f(1)  = 27

So.....average rate of change is

[ f(1) - f (-1) ]

___________  =

     1 - (-1)

[ 27 - 3 ]

______   =

      2

24

__   =

 2

12   ⇒   GOOD JOB!!!

1009 1013 1019 1021 1031

I just realized my calculator was in degree mode so I'm pretty sure that is the reason I was getting the wrong answer

Look at the graph....when x = - 1, then y = -1/4  = -.25

So f(x) =f (-1)  = -.25

Similarly.....when x = 1, then y = -1/2  =  -.50

So  ...f(1) =  -.50

Let the original number of grapes = x

Then the original number of oranges = 1.2x

10% increase in oranges = (1.1)(1.2)x

9% increase in number of grapes = (1/09)x

So.....when126 fruits were added

x + 1.2x + 126  =  (1.1)(1.2)x  + (1.09)x        simplify

2.2x + 126  =  1.32x + 1.09x

2.2x - 1.32x - 1.09 x  =  - 126

-.21x = -126   divide both sides by  -.21

x = 600  =   original number of grapes

1.2(600)  =  720  =  original number of oranges

how did you get this 

f(-1)  =  -.25

f(1)  = -.50

f(-1)  =  -.25

f(1)  = -.50

Avg rate of change =

 [ f(1)  - f (-1) ]

___________   =

      1 - (-1) 

[ -.50  - (-.25) ]

____________   =

          2

-.25

_____   =

   2

-.125

The sample space is not 36.  If we consider order of the dice as important then the size of the sample space is 36

but that is not the sample space.  The sample space consists of all pairs of numbers 1-6, i.e.

(1,1), (1,2) ... (1,6), (2,1), (2,2) ... (6,6)

Usually when rolling dice the order of dice is not considered important so it would be more accurate to describe the sample space

as all pairs of number 1-6 where the first number is greater than or equal to the second number (or vice versa).

In this case the size of the sample space is 6+(6*5)/2 = 21

(1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,1)... (4,4), (5,1) ... (5,5), (6,1)... (6,6)

( x + y)^2  - (x - y)^2

_________________           factor the numerator as a difference of squares

              y 

[ ( x + y) + ( x - y) ] [ (x + y)  - (x - y) ]

_______________________________

                        y

[  2x] [ 2y ]

_________  =

        y

4x =

4(6)  =

24

Avg rate of change =

[ f(-2) - f(-5) ]

___________   =

   -2 -  (-5)

[ (-2)^2 - 4 ] - [ (-5)^2 - 4 ]

_____________________    =

              3

[ 4    -   25 ] 

__________    =

       3

[ -21 ]

_____   =   

    3

-7

NO Max! Even though I wrote the last terms as 100s, but they are added up as follows:

(1*2) + (1*3) + (1*4) +..................+(1*100) - Notice that the first term is 2 NOT 1. For 99 terms. Then it continues...

(2*3) + (2*4) + (2*5) +..................+(2*100) - where the first term is 6 for 98 terms. Then it continues.......

(3*4) + (3*5) + (3*6) +..................+(3*100) - where the first term is 12 for 97 terms. And so on for a total of 99 sequences. Notice the sum of the sequences listed above:

5,049, 10,094, 15132.......etc. They are all correct with  last terms as 100, 200, 300.......9900.

The total I got for 12,582,075 is correct.

NOTE: If you divide your total of 1,508, 082,510 / 99 =15,233,157 per sequence!, which is IMPOSSIBLE!. The biggest total is of the 58th sequence with a total of: 193,662!.

Sorry Melody I didn't know.

check that you've got it set to use whatever units you are, radians or degrees

My first attempts to reply wouldnot go through. I am trying to move one answer up in the list of responses.

My original response:

Thank you for the hint. I am still a bit confused. I am guessing the answer is 40-22=18. Not sure about figuring out the 22 part. Is it a trick question or am I just not "getting it"?