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sqrt ( 9 - [ 3 sin θ ]^2 )    =

sqrt  ( 9  -  9sin^2 θ )  =

sqrt [  9 ( 1 - sin^2 θ ) ]  =

sqrt ( 9 * cos^2  θ )  =

sqrt (9)  *  sqrt ( cos^2 θ )  =

3 cos θ

I think it's  "C"

Reasoning :

"Free responses"    might encourage lengthy  answers which are usually more time-consuming  for the respondent than multiple-choice questions....thus....fewer questionnaires might be completed on time

Yeah I got that one, it was easier than the other one

Good job,  RP  !!!!

Both are correct  !!!

Looks like  "B"  is the only disadvantage, NSS

OK...glad to help  .....!!!!

The second one is easier, RP

√ [5 p^3 ]

_________     =    (1/3) √ [   (5 p^3)  /  (2p^3) ]    

3 √ [2 p^3 ]

The p^3's   will "cancel"     and we are left with

(1/3) √ [ 5 / 2 ]  =      

√ 5

_____            (do you see this  ??? )

3 √2

We want to get rid of the radical in the denominator  [ this is called "rationalizing the denominator" ]

We can do this in this manner.....Multiply top/bottom by √2     and note that √2 *√2  = √4   = 2

√5 * √2

__________

3 √2 * √2

√10

_____

3 * 2

√10

_____

  6

Thank you.  It makes sense now that I see it.

Agreed........I'll try to answer your second problem with extensive explanations

oh good cause i wasn't sure it was that or if it was A

9 / (1 + cos x  )    + 9 / ( 1 - cos x)

We can do this

9 ( 1 - cos x)                            9 ( 1 + cos x)

_________________     +    ___________________

(1 + cos x) (1 - cos x)             (1 - cos x) ( 1 + cos x)

9 - 9 cos x  +  9  +  9 cos x 

______________________

   1  - cos^2 x

   18

_______

 sin^2 x

  18   *            1

               _______   

               (sin x)^2

18  *    [  1  / sin x ] ^2  

18  *  [ csc x ] ^2

18 csc^2 x

You are right once again, good job!

That is a lot to process...

I DID look at the graph to make sure that I gave you the correct answers......but...it is also helpful to think about WHY  the graph looks as it does.....

Yeah I know.....we are using a lot of   exponential rules here

Here are a few that I employed

^m √ [ a^n ]    converts  to exponential form   (a)^(n/m)

And

a^(m)  *  a^(n)  =   a^(m + n)

And

Remember.... we can take  something out of a radical  when  we have  (a)^(n/m)  and n  ≥ m

For instance...let's suppose that we have     ³ √ [ 4^5]

In  exponential form....we have  (4)^(5/3)

To  determine how many powers of 4 come out of the radical and how many stay in.....write the exponent as a mixed number

4^(1 +2/3)

The whole number tells us that 1 power of 4  comes out of the radical  and  the numerator of the fraction tells us that 2 powers of 4 stay inside the radical...so we have

4 ³√ [4^2 ]    =    4 ³√ [ 16 ]

But note that  16  =  2^4....so...we can apply this rule once more

We  have    4 ³√ [2^4 ]    =   4 (2)^(4/3)

To determine how many powers of 2 that we can take out of the radical and how many we can leave in...write this as

4 (2)^(1 + 1/3)....so....we can take 1 power of 2 out of the radical and 1 power of 2 stays in

So..putting this all together, we have

(4 * 2) ³√2     =   8 ³√2

Hang in there, RP......you will begin to understand this

Thank you for your answer.

Did you use any graphic calculator to find the answers or did you figure it out any other way? :)

Thanks again, it's really helpful.

You are correct ^-^

\(x^4 -x^3 - 2x - 4 = (x^4-4)-(x^3+2x) = \\ (x^2-2)(x^2+2) - x(x^2+2) = \\ (x^2+2)(x^2-x-2)=\\ (x+\sqrt{2}i)(x-\sqrt{2}i)(x-2)(x+1)\)

2) A fifth degree polynomial will have 5 complex zeros

3) \(x^4+2x^3-16x^2-2x+15 = \\ (x^4-16x^2+15) +(2x^3-2x) =\\ (x^2-15)(x^2-1)+2x(x^2-1) = \\ (x^2+2x-15)(x^2-1) = \\ (x-3)(x+5)(x-1)(x+1) \\ \\ \text{and the zeros can be read off as }\\ \\ x=3,~-5,~1,~-1\)

y  =    1  / [ (x + 1) (x - 1) ]  =   1 / ( x^2  - 1 )

The  vertical asymptotes  are the x values that make the denominator  = 0

These are the lines x  = -1   and x  = 1

Since we have a lower degree polynomial over a higher degree polynomial...we will have a horizontal asymptote of 0   [ this will always be true in a  lower / higher   situation ]

The range is a little difficult

Note that when x is 0, the function will reach its maximum negative value  of -1

When  x  is either a large negative  or positive value   1 / [ x^2  - 1 ]  will be almost  0

As x  moves  closer  to the left   vertical  asymptote  x  = -1  from the left side  or  closer to the right vertical asymptote x = 1 from the right side...the function aprroaches infinity

For example.....let x  =   -1.0001   or   1.0001  and we get the large positive value  of ≈ 5000

And this will increase as we get closer to each of these asymptotes in this manner

As the function moves closer to the left asymptote of x  = -1 form the right or closer to  the right vertical asymptote of x = 1 from the left, the function approaches negative infinity

For example when  x  =  -.9999  or   x  =.9999 the function  value   ≈ -5000

And this will decrease as we get closer to these aymptotes in this manner

So....the range is  ( -infinity , 1 ] U  (0 , infinity )

What about the 2? Where does that go?

Thanks for helping! I still don't quite understand though, there are a lot of steps and it's confusing >.<

\(\dfrac{\sqrt[3]{4r^3}}{\sqrt[3]{9r}} = \dfrac{r\sqrt[3]{4}}{\sqrt[3]{9r}} \\ \\ \text{This is really as far as it can be simplified}\\ \text{but it's common to want the denominator to be free of radicals}\\ \dfrac{r\sqrt[3]{4}}{\sqrt[3]{9r}} =\dfrac{r\sqrt[3]{4}}{\sqrt[3]{9r}} \dfrac{\sqrt[3]{9^2r^2}}{\sqrt[3]{9^2r^2}} = \dfrac{r\sqrt[3]{324r^2}}{\sqrt[3]{9^3r^3}}=\dfrac{r\sqrt[3]{324r^2}}{9r}= \dfrac{\sqrt[3]{324r^2}}{9}\)

First one.....convert to exponential form

 5 ( 4n)^(1/4)

__________        

2  (27n)^(1/4)

We can "cancel" the n's and get

5 (4)^(1/4)

_________         multiply top/bottom by   (27)^(3/4)

2 (27)^(1/4)

5 ( 4 )^(1/4) (27)^(3/4)

__________________

2 (27)^(1/4) 27^(3/4)

5 [ 4 (27)^3 ] ^(1/4)

_________________

  2 * 27

5 [ 4 (3^3)^3 ] ^(1/4)

________________

   54

5 [ 4 * 3^9 ]^(1/4)

______________        

    54

Note :  (3^9)^(1/4)  =  3^(9/4)  =  3^(2 + 1/4)...we can take 3^2  out of the radical  and leave one power of 3 inside

5 * 3^2   [ 4 * 3 ]^(1/4)

_________________

    54

5 * 9 [ 12]^(1/4)

_____________

54

5 [12]^(1/4)

_________

     6

5 ⁴√12

______

    6

Second one only

Add up all the numbers: 1+1.5+8+9.2+6.8+4.5+4.8+2.5+0.76= 39.06 and divide by how many numbers there are: 39.06/9=4.34

Group A= 4.34

Group B= 2.17

So, Group A is double the value for Group B. You are correct! Great job~